# Symmetric Lie algebras and Hermitian symmetric spaces

A symmetric space carries both a connection and an algebra of infinitesimal motions. An invariant complex structure links these two descriptions: its parallelism becomes an inner derivation, generated by a central isotropy element. We first establish this link and the decomposition needed to interpret its curvature.

All manifolds are smooth, Hausdorff and second countable. Spaces and acting groups are connected unless stated otherwise. All Lie algebras are finite dimensional and real, except where complexification is explicit. The curvature convention is
\[
R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.
\]
A symmetric presentation means
\[
(G^\sigma)^0\subseteq H\subseteq G^\sigma,\qquad
\mathfrak g=\mathfrak h\oplus\mathfrak m,
\]
where \(\sigma\) is an involution and the summands are the positive and negative eigenspaces of its differential \(\theta\). The canonical connection and all quotient conventions are those proved in [Symmetric spaces B.1–B.2](symmetric-spaces.md#lemma-b-1). In particular,
\[
R(X,Y)Z=-[[X,Y],Z]\quad(X,Y,Z\in\mathfrak m).
\tag{0.1}
\]

## A. Complex structures preserved by reflections

A complex structure is a smooth endomorphism \(J\) with \(J^2=-I\) that is induced by a complex coordinate atlas. A smooth map between complex manifolds is holomorphic when its differential is complex linear in these charts: real differentiability with complex-linear derivative is precisely complex differentiability. A Hermitian metric is a Riemannian metric with \(g(JX,JY)=g(X,Y)\). Put \(\omega(X,Y)=g(JX,Y)\).

**Lemma A.1 (The two Kähler conditions).** For a Hermitian metric on a complex manifold,
\[
d\omega=0\quad\Longleftrightarrow\quad\nabla J=0,
\tag{A.1}
\]
where \(\nabla\) is the Levi-Civita connection. Either condition defines a Kähler metric.

**Proof.** If \(J\) is parallel, metric compatibility makes \(\omega\) parallel. For a two-form and a torsion-free connection, expansion of the definition of the exterior derivative, replacing brackets by differences of covariant derivatives, gives
\[
d\omega(X,Y,Z)=
(\nabla_X\omega)(Y,Z)+(\nabla_Y\omega)(Z,X)+(\nabla_Z\omega)(X,Y).
\]
Thus \(d\omega=0\).

Conversely work in complex coordinates \(z^1,\ldots,z^k\), and extend the metric and connection complex bilinearly. Write
\[
\partial_a=\tfrac12(\partial_{x^a}-i\partial_{y^a}),\qquad
\partial_{\bar a}=\tfrac12(\partial_{x^a}+i\partial_{y^a}).
\]
These fields commute. Hermitian compatibility gives
\(g_{ab}=g_{\bar a\bar b}=0\), while the matrix \(g_{a\bar b}\) is nondegenerate. Since \(J\partial_a=i\partial_a\) and \(J\partial_{\bar a}=-i\partial_{\bar a}\), closure of \(\omega\), evaluated on \(\partial_a,\partial_b,\partial_{\bar c}\) and their conjugates, gives
\[
\partial_a g_{b\bar c}=\partial_b g_{a\bar c},\qquad
\partial_{\bar a}g_{c\bar b}=\partial_{\bar b}g_{c\bar a}.
\tag{A.2}
\]
The complex-bilinear extension of the Koszul formula proved in [Symmetric spaces A.1](symmetric-spaces.md#theorem-a-1) now yields
\[
\begin{aligned}
2g(\nabla_{\partial_{\bar c}}\partial_a,\partial_b)
 &=\partial_a g_{b\bar c}-\partial_b g_{a\bar c}=0,\\
2g(\nabla_{\partial_{\bar c}}\partial_a,\partial_{\bar b})
 &=\partial_{\bar c}g_{a\bar b}-\partial_{\bar b}g_{a\bar c}=0.
\end{aligned}
\]
Hence the mixed derivative is zero. Also
\(g(\nabla_{\partial_c}\partial_a,\partial_b)=0\), because every term in its Koszul formula involves a zero pure-type metric coefficient. Nondegeneracy of the mixed matrix says that \(\nabla_{\partial_c}\partial_a\) has type \((1,0)\). Conjugation gives the corresponding assertions for type \((0,1)\). The connection preserves both eigensubbundles of \(J\), which is exactly \(\nabla J=0\). These are coordinate identities and therefore hold on the manifold. □

**Theorem A.2 (Hermitian symmetry and parallelism).** A Hermitian symmetric space is Kähler. Conversely, a Kähler structure on a connected Riemannian symmetric space makes every point symmetry holomorphic.

**Proof.** A Hermitian symmetric space has, at every point \(p\), a holomorphic isometry \(s_p\) fixing \(p\) with derivative \(-I\). An isometry preserves the Levi-Civita connection by [Riemannian connections A.3](riemannian-connections-and-convex-neighbourhoods.md#theorem-a-3), and holomorphicity preserves \(J\). It therefore preserves the tensor
\((\nabla_XJ)Y\), by the tensor-derivative rule in [Linear connections C.1](linear-and-affine-connections.md#theorem-c-1). At \(p\), its two inputs acquire two minus signs and its output one. Thus
\[
-(\nabla_XJ)Y=(\nabla_{-X}J)(-Y)=(\nabla_XJ)Y,
\]
so \(\nabla J=0\) everywhere. Lemma A.1 gives the first conclusion.

Conversely, a point symmetry is an isometry and hence affine. Its pullback of the parallel tensor \(J\) is parallel. At its centre this pullback equals \(J\), since \(-I\) commutes with \(J\). Two parallel tensors agreeing at one point agree along every piecewise smooth path: their components in parallel frames satisfy the same linear ODE, as in [Linear connections C.1](linear-and-affine-connections.md#theorem-c-1) and [Connections D.2](connections-and-parallel-transport.md#theorem-d-2). The set reachable from a point by piecewise smooth paths is open, and its complement is open, since coordinate balls join their points by straight coordinate segments. Connectedness makes every point reachable. The pullback is therefore \(J\) everywhere. Its differential is complex linear, which proves holomorphicity. □

**Theorem A.3 (The full stabilizer condition).** On a symmetric presentation, an endomorphism \(I\) of \(\mathfrak m\) determines a \(G\)-invariant complex structure exactly when
\[
I^2=-I_{\mathfrak m},\qquad
I\operatorname{Ad}(h)|_{\mathfrak m}
=\operatorname{Ad}(h)|_{\mathfrak m}I\quad(h\in H).
\tag{A.3}
\]
Here \(I_{\mathfrak m}\) denotes the identity operator. Every positive \(H\)-invariant inner product compatible with this complex structure gives a complete Hermitian symmetric Kähler metric. Given a positive invariant inner product \(b\), a compatible one is
\[
b_I(X,Y)=\tfrac12\bigl(b(X,Y)+b(IX,IY)\bigr).
\tag{A.4}
\]

**Proof.** Translating \(I\) from the origin defines a tensor precisely when changing a coset representative by \(h\in H\) has no effect. This is the second condition in (A.3), by the associated-tensor construction in [Homogeneous spaces C.1](homogeneous-spaces-and-invariant-connections.md#theorem-c-1). The first condition is the almost complex condition.

Every bracket of two vectors in \(\mathfrak m\) lies in \(\mathfrak h\). Consequently all four \(\mathfrak m\)-bracket terms in [Homogeneous spaces G.4](homogeneous-spaces-and-invariant-connections.md#theorem-g-4) vanish. That theorem constructs complex coordinates for the invariant almost complex structure, with a complete analytic Frobenius proof. It also proves that the \(G\)-action is holomorphic. This establishes sufficiency for integrability without requiring a separate unproved integrability theorem. Necessity follows by evaluating an invariant complex structure at the origin.

The form (A.4) is positive, symmetric and invariant, and replacing both arguments by their \(I\)-images leaves it unchanged. A compatible invariant metric has the canonical Levi-Civita connection by [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2). The same theorem makes every invariant tensor, in particular \(J\), parallel and makes the connection complete. Its global point symmetries are isometries; Theorem A.2 makes them holomorphic, and Lemma A.1 makes the metric Kähler. Every condition has been imposed on the full \(H\), including any disconnected components. □

## B. Solvable ideals and traces

Write \(B(X,Y)=\operatorname{tr}_{\mathfrak g}(\operatorname{ad}X\,\operatorname{ad}Y)\) for the Killing form and \(\mathfrak r\) for the solvable radical. Solvability, radicals and finite-dimensional representations were developed with proofs in [Symmetric spaces H.2–H.6](symmetric-spaces.md#lemma-h-2).

**Lemma B.1 (A solvable ideal on an irreducible module).** If \(\mathfrak a\) is a solvable ideal in a complex Lie algebra \(\mathfrak g\), then on any nonzero finite-dimensional irreducible complex \(\mathfrak g\)-module \(V\), every element of \(\mathfrak a\) acts by a scalar and \([\mathfrak g,\mathfrak a]\) acts by zero.

**Proof.** Lie's theorem, [Symmetric spaces H.3](symmetric-spaces.md#theorem-h-3), supplies a nonzero vector \(v\) and a linear functional \(\lambda\) on \(\mathfrak a\) with \(yv=\lambda(y)v\). There are only finitely many possible functionals of common eigenvectors: triangularize the action of \(\mathfrak a\); the last nonzero coordinate of any common eigenvector identifies its functional with one of the finitely many diagonal functionals.

Fix \(x\in\mathfrak g\). The ideal \(\mathfrak a\) is stable under \(\operatorname{ad}x\) and its exponential. The finite-dimensional matrix exponential identity
\[
e^{t\rho(x)}\rho(y)e^{-t\rho(x)}
=\rho(e^{t\operatorname{ad}x}y)
\tag{B.1}
\]
follows by differentiating both sides; they solve the same linear commutator ODE with the same initial value, using [Local tools 2.3](local-tools-for-bundles-and-transport.md#2-differential-equations-and-their-parameters). Thus \(e^{t\rho(x)}v\) is a common eigenvector for \(\mathfrak a\), with functional
\(\lambda\circ e^{-t\operatorname{ad}x}\). For real \(t\) this is a continuous map into the finite set of diagonal functionals. It is constant, since an interval is connected. Differentiating at zero gives
\(\lambda([x,y])=0\) for every \(y\in\mathfrak a\).

Let \(W\) be the common eigenspace for \(\lambda\), containing \(v\). For \(w\in W\),
\[
\rho(y)\rho(x)w
=\rho(x)\rho(y)w+\rho([y,x])w
=\lambda(y)\rho(x)w.
\]
The last equality uses \([y,x]\in\mathfrak a\) and its zero \(\lambda\)-value. Thus \(W\) is \(\mathfrak g\)-invariant. Irreducibility gives \(W=V\), proving the scalar assertion and the vanishing of the commutators. □

**Theorem B.2 (The radical from the Killing form).** For every real or complex Lie algebra,
\[
\mathfrak r
=\{X\in\mathfrak g:B(X,[\mathfrak g,\mathfrak g])=0\}.
\tag{B.2}
\]
More generally, in any finite-dimensional representation \(\rho\), a solvable ideal \(\mathfrak a\) satisfies
\[
\operatorname{tr}\bigl(\rho(X)\rho(Y)\bigr)=0
\quad(X\in\mathfrak a,\ Y\in[\mathfrak g,\mathfrak g]).
\tag{B.3}
\]

**Proof.** Over \(\mathbb C\), choose a finite invariant filtration of the module with irreducible quotients. It exists by choosing a nonzero invariant subspace of least dimension, then proceeding in the quotient; dimension decreases at every step. On each irreducible quotient, Lemma B.1 makes \(\rho(X)\) a scalar. The trace of \(\rho(Y)\) there is zero because \(Y\) is a sum of brackets, each represented by a matrix commutator. Hence the trace of their product is zero. A basis adapted to the filtration makes both operators block upper triangular, so the trace of their product is the sum of these quotient traces. This proves (B.3). For real algebras and modules, complexify the matrices and the solvable ideal. Solvability is preserved by complexification by [Symmetric spaces H.2](symmetric-spaces.md#lemma-h-2), and matrix traces are unchanged; the same conclusion follows.

Apply (B.3) to the adjoint representation and \(\mathfrak a=\mathfrak r\). This proves one inclusion in (B.2). Denote the space on its right by \(\mathfrak b\). Killing-form invariance and the fact that the derived algebra is an ideal give
\[
B([Z,X],Y)=-B(X,[Z,Y])=0
\quad(X\in\mathfrak b,\ Y\in[\mathfrak g,\mathfrak g]).
\]
Thus \(\mathfrak b\) is an ideal. For any ideal, restriction of the ambient Killing form to it equals its own Killing form, as proved in [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5). Since
\([\mathfrak b,\mathfrak b]\subseteq[\mathfrak g,\mathfrak g]\), we have
\[
B_{\mathfrak b}(\mathfrak b,[\mathfrak b,\mathfrak b])=0.
\]
The solvability criterion in that same earlier theorem makes \(\mathfrak b\) solvable. Maximality of the radical gives \(\mathfrak b\subseteq\mathfrak r\), proving equality. □

## C. Orthogonal symmetric algebras

A symmetric Lie algebra is a real Lie algebra with an involution \(\theta\), whose eigenspaces we denote by \(\mathfrak h,\mathfrak m\). Thus
\[
[\mathfrak h,\mathfrak h]\subseteq\mathfrak h,\qquad
[\mathfrak h,\mathfrak m]\subseteq\mathfrak m,\qquad
[\mathfrak m,\mathfrak m]\subseteq\mathfrak h.
\]
It is **orthogonal** here if some positive inner product on \(\mathfrak g\) makes every \(\operatorname{ad}A\), \(A\in\mathfrak h\), skew-adjoint. In particular this condition holds when the connected linear group generated by these adjoint operators is compact: average a positive form using [Symmetric spaces D.1](symmetric-spaces.md#lemma-d-1). Averaging the form with its transform under \(\theta\) also makes the involution orthogonal. The arguments below require only this invariant-form condition, so they also apply when that generated group merely has compact closure.

The pair is **effective** if no nonzero ideal lies in \(\mathfrak h\). It has **Euclidean type** if it is effective and \([\mathfrak m,\mathfrak m]=0\). A semisimple pair has **compact type** when \(B\) is negative definite on \(\mathfrak g\). An effective semisimple pair has **noncompact type** when
\[
B_\theta(X,Y)=-B(X,\theta Y)
\tag{C.1}
\]
is positive definite. A pair with zero algebra is allowed as an empty summand.

**Lemma C.1 (Faithfulness and the even Killing form).** The kernel of the action of \(\mathfrak h\) on \(\mathfrak m\) is exactly the largest ideal contained in \(\mathfrak h\). Thus effectiveness is equivalent to faithfulness of that action. On an orthogonal pair,
\[
B(\mathfrak h,\mathfrak m)=0,\qquad
B(A,A)=-\|\operatorname{ad}A\|_{\mathrm{HS}}^2
\quad(A\in\mathfrak h),
\tag{C.2}
\]
where the norm is taken for an invariant positive form on \(\mathfrak g\). In particular \(B|_{\mathfrak h}\) is negative definite if
\(\mathfrak h\cap Z(\mathfrak g)=0\), and hence if the pair is effective. Quotienting by the isotropy kernel gives an effective orthogonal symmetric algebra with the same odd tangent space and canonical curvature.

**Proof.** Let \(\mathfrak k\) be the kernel in question. For \(A\in\mathfrak k\), \(D\in\mathfrak h\), \(X\in\mathfrak m\), Jacobi gives
\([[D,A],X]=[D,[A,X]]-[A,[D,X]]=0\).
Thus \([\mathfrak h,\mathfrak k]\subseteq\mathfrak k\), and by definition
\([\mathfrak m,\mathfrak k]=0\). This makes \(\mathfrak k\) an ideal. Conversely an ideal contained in \(\mathfrak h\) brackets with \(\mathfrak m\) into both eigenspaces, hence into zero, so is contained in \(\mathfrak k\).

Every automorphism preserves the Killing form, since it conjugates adjoint operators. Applying \(\theta\) to one even and one odd argument reverses their pairing, proving the first equality of (C.2). In an orthonormal basis a skew matrix \(T\) satisfies
\(\operatorname{tr}T^2=-\operatorname{tr}(T^TT)=-\sum_{i,j}T_{ij}^2\).
Take \(T=\operatorname{ad}A\). Equality to zero means that \(A\) is central. The stated intersection condition therefore gives negative definiteness. In an effective pair any central even line is an ideal in \(\mathfrak h\), so that intersection is zero.

For this last assertion, the ideal \(\mathfrak k\) is even, so \(\theta\) descends to \(\mathfrak g/\mathfrak k\), with eigenspaces \(\mathfrak h/\mathfrak k\) and the unchanged \(\mathfrak m\). Its even action is faithful: an even class annihilating every odd vector has a representative in \(\mathfrak k\), by its definition, and is therefore zero. The preceding kernel characterization proves effectiveness. An invariant positive form descends as the quotient inner product: identify the quotient with the orthogonal complement of \(\mathfrak k\), which is stable under every even skew-adjoint operator because \(\mathfrak k\) is invariant. These induced operators are still skew-adjoint. Finally the quotient map is the identity on the odd space and preserves brackets; its even kernel acts trivially there. Thus the expression \(-[[X,Y],Z]\) in (0.1) is unchanged for all odd \(X,Y,Z\). This proves the tangent and curvature assertions without assuming a global group quotient. □

**Theorem C.2 (Decomposition by the Killing operator).** An effective orthogonal symmetric Lie algebra is a direct sum of symmetric ideals
\[
\mathfrak g=\mathfrak g_0\oplus\mathfrak g_c\oplus\mathfrak g_n.
\tag{C.3}
\]
The first is a Euclidean pair
\(\mathfrak g_0=\mathfrak h_0\ltimes\mathfrak m_0\) with faithful skew-adjoint isotropy. The second and third are semisimple of compact and noncompact type. Each semisimple part splits further into effective symmetric ideals with irreducible nonzero odd representation. If \(\mathfrak m\) generates \(\mathfrak g\), then \(\mathfrak h_0=0\) and the Euclidean ideal consists only of translations.

**Proof.** Restrict an invariant positive form to \(\mathfrak m\), writing it as \(\langle\, ,\,\rangle\). Define its self-adjoint Killing operator \(S\) by
\[
B(X,Y)=\langle SX,Y\rangle\quad(X,Y\in\mathfrak m).
\]
Invariance of both forms makes \(S\) commute with every \(\operatorname{ad}A|_{\mathfrak m}\). The real spectral theorem proved in [De Rham decomposition H.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-h-2) gives an orthogonal splitting into invariant eigenspaces
\(\mathfrak m=\bigoplus_\lambda V_\lambda\).
The corresponding Killing forms are \(\lambda\langle\, ,\,\rangle\).

For distinct eigenvalues, \(X\in V_\lambda,Y\in V_\mu\) and \(A\in\mathfrak h\), Killing invariance gives
\[
B([X,Y],A)=B(X,[Y,A])=0,
\]
since \([Y,A]\in V_\mu\). Nondegeneracy of \(B|_{\mathfrak h}\), from C.1, proves
\([V_\lambda,V_\mu]=0\). The same calculation, now using \(B(V_0,\mathfrak m)=0\), proves
\[
[V_0,\mathfrak m]=0.
\tag{C.4}
\]

For \(\lambda\ne0\), set \(\mathfrak h_\lambda=[V_\lambda,V_\lambda]\). Jacobi shows that it is an ideal in \(\mathfrak h\), preserves \(V_\lambda\), and acts trivially on every other \(V_\mu\). Distinct \(\mathfrak h_\lambda\)'s commute: expand the bracket of an element of \([V_\lambda,V_\lambda]\) with one of \([V_\mu,V_\mu]\), and use the vanishing cross brackets. They have zero intersections and independent sum, since an element in the intersection of one with the sum of the others acts trivially on all of \(\mathfrak m\), so is zero by faithfulness.

They are also pairwise \(B\)-orthogonal: for \(A\in\mathfrak h_\mu\),
\(B([X,Y],A)=B(X,[Y,A])=0\) when \(X,Y\in V_\lambda\), \(\lambda\ne\mu\).
Let \(\mathfrak h_0\) be their orthogonal complement inside the negative-definite space \((\mathfrak h,B)\). Invariance makes it an ideal of \(\mathfrak h\). Orthogonal complementary ideals commute, because their bracket belongs to both. For \(A\in\mathfrak h_0\), \(X,Y\in V_\lambda\), \(\lambda\ne0\),
\[
B([A,X],Y)=B(A,[X,Y])=0.
\]
The action preserves \(V_\lambda\), where \(B\) is nondegenerate, so
\([\mathfrak h_0,V_\lambda]=0\).

Consequently
\[
\mathfrak g_\lambda=\mathfrak h_\lambda\oplus V_\lambda
\quad(\lambda\ne0),\qquad
\mathfrak g_0=\mathfrak h_0\oplus V_0
\]
are mutually commuting, involution-stable ideals whose sum is all of \(\mathfrak g\). Equation (C.4) makes \(\mathfrak g_0\) the asserted semidirect product; its isotropy is faithful since an element acting trivially on \(V_0\) acts trivially on every summand.

On \(\mathfrak g_\lambda\), the Killing form is the restriction of \(B\), either by the ideal formula of [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) or by the commuting direct-sum matrices. It is nondegenerate: its even part is negative definite, its odd part is the nonzero multiple \(\lambda\langle\, ,\,\rangle\), and these parts are orthogonal. That earlier Killing criterion makes \(\mathfrak g_\lambda\) semisimple. It has compact type if \(\lambda<0\), and noncompact type if \(\lambda>0\), by (C.1).

For the final irreducibility statement, decompose any \(V_\lambda\), \(\lambda\ne0\), into irreducible \(\mathfrak h\)-invariant real subspaces. This needs no general representation theorem: an invariant subspace has invariant orthogonal complement because the action is skew, so induction on dimension supplies the decomposition. For two orthogonal invariant pieces \(V,W\) of the same eigenspace, the calculation
\(B([V,W],A)=B(V,[W,A])=0\) again gives \([V,W]=0\). Repeating the preceding bracket argument splits \(\mathfrak g_\lambda\) into the ideals \([V,V]\oplus V\). There is no residual even part because \(\mathfrak h_\lambda=[V_\lambda,V_\lambda]\). Each such piece is effective, and its odd representation is irreducible by construction. It is nontrivial: if \([\,\mathfrak h,V\,]=0\), the same Killing pairing would give \([V,\mathfrak m]=0\), so \(V\) would be central, contradicting the nondegeneracy on that piece.

Finally the subalgebra generated by \(\mathfrak m\) is \([\mathfrak m,\mathfrak m]\oplus\mathfrak m\): Jacobi and the symmetric bracket rules make this a subalgebra. Its even part is the sum of the \(\mathfrak h_\lambda\), \(\lambda\ne0\). If it is all of \(\mathfrak g\), its complementary \(\mathfrak h_0\) must vanish. □

**Corollary C.3 (The central-kernel hypothesis).** The decomposition into Euclidean, compact and noncompact symmetric ideals also holds for an orthogonal pair satisfying
\[
\mathfrak h\cap Z(\mathfrak g)=0.
\tag{C.5}
\]
The compact ideal may additionally contain semisimple ideals entirely in \(\mathfrak h\), contributing no tangent directions. Removing those ideals leaves the effective decomposition of C.2.

**Proof.** Let \(\mathfrak k\) be the even ideal from C.1. Restrict the invariant positive form on \(\mathfrak g\) to \(\mathfrak h\). The orthogonal complement of \(\mathfrak k\) in \(\mathfrak h\) is an ideal, by invariance, and the two ideals commute. If \(Z\) lies in the centre of \(\mathfrak k\), it therefore commutes with all of \(\mathfrak h\); it also commutes with \(\mathfrak m\) by the definition of \(\mathfrak k\). Condition (C.5) gives \(Z=0\).

The adjoint operators of \(\mathfrak k\) on itself are skew and its centre is zero. The matrix calculation in C.1 makes its own Killing form negative definite. By [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) it is semisimple. Its Killing form equals \(B|_{\mathfrak k}\), because it is an ideal of \(\mathfrak g\). Thus its \(B\)-orthogonal complement in \(\mathfrak g\) is an ideal complementary to it: solving the nondegenerate pairing on \(\mathfrak k\) gives the direct sum, even if \(B\) is degenerate elsewhere. The ideals commute and are involution stable. Every odd vector belongs to this complement by (C.2).

The complementary pair is effective. An even element in its isotropy kernel would belong to \(\mathfrak k\) by C.1 and also to its complementary ideal, hence be zero. Apply C.2 to it, and add the compact, purely even ideal \(\mathfrak k\) to the compact summand. □

**Proposition C.4 (Realizing a semisimple pair).** Every semisimple orthogonal symmetric Lie algebra has a connected Lie-group presentation with closed connected isotropy and the prescribed Lie algebra and involution.

**Proof.** Inside \(GL(\mathfrak g)\), let
\[
\operatorname{Aut}(\mathfrak g)
=\{F:F[X,Y]=[FX,FY]\text{ for all }X,Y\}.
\]
Checking on a basis expresses this set by finitely many polynomial equations, so it is closed relative to \(GL(\mathfrak g)\). Invariant connections A.1 makes it an embedded Lie group. Differentiation of its equations shows that its Lie algebra lies in \(\operatorname{Der}(\mathfrak g)\). Conversely a derivation \(D\) satisfies
\[
\frac{d}{dt}\left(e^{-tD}[e^{tD}X,e^{tD}Y]\right)=0
\]
by the derivation identity. Hence \(e^{tD}\) is an automorphism, proving equality of these Lie algebras. [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6) identifies \(\operatorname{Der}(\mathfrak g)\) with \(\operatorname{ad}\mathfrak g\), faithfully since the centre is zero.

Take \(G=\operatorname{Aut}(\mathfrak g)^0\) and let its involution be \(F\mapsto\theta F\theta^{-1}\). Under the adjoint identification its differential is \(\theta\). The identity component \(H\) of its fixed subgroup is closed and has Lie algebra \(\mathfrak h\), by [Symmetric spaces B.1](symmetric-spaces.md#lemma-b-1) and the closedness of components. Its action preserves the given positive form: each \(\exp(\operatorname{ad}A)\), \(A\in\mathfrak h\), preserves it, and these exponentials generate connected \(H\), by [Flat connections A.1](flat-connections-and-infinitesimal-holonomy.md#lemma-a-1). Thus this is a symmetric presentation with a positive invariant tangent metric. □

## D. Metrics, curvature and the dual pair

**Theorem D.1 (Every invariant metric and the Ricci trace).** For an effective semisimple orthogonal pair, let
\(\mathfrak g=\bigoplus_a\mathfrak g_a\) be its decomposition from C.2 into irreducible symmetric ideals, with odd spaces \(V_a\). Every positive infinitesimally invariant inner product on \(\mathfrak m\) is an orthogonal sum
\[
b|_{V_a}=c_a\epsilon_a B|_{V_a},\qquad c_a>0,
\tag{D.1}
\]
where \(\epsilon_a=-1\) on compact factors and \(+1\) on noncompact factors. For an actual \(G/H\), full \(H\)-invariance must also hold; in particular, components permuted by \(H\) must have the corresponding equal scales.

For its canonical metric connection,
\[
\operatorname{Ric}(X,Y)=-\tfrac12 B(X,Y).
\tag{D.2}
\]
On one irreducible summand write \(B=\lambda b\), with \(\lambda\ne0\). Then
\[
b(R(X,Y)Y,X)=\lambda^{-1}B([X,Y],[X,Y])
\quad(X,Y\in V_a).
\tag{D.3}
\]
Compact factors have nonnegative sectional and positive Ricci curvature; noncompact factors have nonpositive sectional and negative Ricci curvature.

**Proof.** Fix one reference invariant positive form and its irreducible splitting from C.2. The Lie algebra \(\mathfrak h_a=[V_a,V_a]\) acts trivially on \(V_b\) for \(b\ne a\), and nontrivially on \(V_a\). The span \([\mathfrak h_a,V_a]\) is invariant and nonzero, so irreducibility makes it all of \(V_a\). For any other invariant bilinear form \(b\),
\[
b([A,X],Y)=-b(X,[A,Y])=0
\quad(A\in\mathfrak h_a,\ X\in V_a,\ Y\in V_b).
\]
Thus the distinct summands are orthogonal for \(b\) too.

On \(V_a\), the self-adjoint operator comparing two positive invariant forms commutes with the isotropy action. Every eigenspace is invariant; irreducibility and [De Rham decomposition H.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-h-2) make this operator scalar. Applying this to the positive definite form \(\epsilon_a B\) proves (D.1). An automorphism in the full stabilizer preserves \(B\) and sends each irreducible ideal to an ideal of the same type, so its invariance imposes the stated equality when it permutes factors.

For the trace computation, \(\operatorname{ad}X\) exchanges \(\mathfrak h\) and \(\mathfrak m\) when \(X\in\mathfrak m\). Put
\(a_X=\operatorname{ad}X|_{\mathfrak h}:\mathfrak h\to\mathfrak m\) and
\(b_X=\operatorname{ad}X|_{\mathfrak m}:\mathfrak m\to\mathfrak h\).
The entry-sum identity for rectangular matrices gives
\(\operatorname{tr}_{\mathfrak h}(b_Xa_Y)=\operatorname{tr}_{\mathfrak m}(a_Yb_X)\).
Also
\[
\operatorname{tr}_{\mathfrak m}(a_Xb_Y-a_Yb_X)
=\operatorname{tr}_{\mathfrak m}\operatorname{ad}[X,Y]=0,
\]
because the even action is skew-adjoint. Hence
\[
B(X,Y)=2\operatorname{tr}_{\mathfrak m}(a_Yb_X).
\]
Formula (0.1) gives
\(R(Z,X)Y=-a_Yb_XZ\). Its trace over \(Z\) is the definition of Ricci, as in [Sectional curvature A.1](sectional-curvature-and-space-forms.md#theorem-a-1). This proves (D.2).

Finally Killing invariance yields
\[
\begin{aligned}
\lambda b(R(X,Y)Y,X)
 &=B(-[[X,Y],Y],X)\\
 &=B([X,Y],[X,Y]),
\end{aligned}
\]
proving (D.3). The bracket is even, where \(B\) is negative definite by C.1. Thus its sign is nonnegative when \(\lambda<0\) and nonpositive when \(\lambda>0\). On a sum of ideals, cross brackets and cross curvature vanish, so the numerator is the sum of those on the individual factors. The Ricci signs follow directly from (D.2) and definiteness of \(B\) on the appropriate odd summands. □

**Theorem D.2 (Global geometry of the two types).** A connected Riemannian symmetric space with effective orthogonal algebra of compact type is compact and has finite fundamental group. One of noncompact type is simply connected, and every exponential map is a diffeomorphism from its tangent space onto the space. These assertions allow reducible spaces and every invariant positive metric.

**Proof.** [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves completeness and homogeneity, and D.1 gives the curvature signs and definite Ricci tensors. In compact type, the finite positive scales in (D.1) give a uniform constant \(c>0\) with
\(\operatorname{Ric}\geq c\,b\): on factor \(a\), this constant is \(1/(2c_a)\), so take the least of these finitely many numbers. For positive dimension \(n\), necessarily \(n\geq2\), since the curvature and Ricci of a one-dimensional manifold vanish by skewness of the curvature tensor, contradicting positive Ricci. [Comparison D.1](comparison-theorems-cut-loci-and-curvature-and-topology.md#theorem-d-1) applies with \(k=c/(n-1)\), proving compactness and finite fundamental group.

In noncompact type, D.1 gives nonpositive sectional curvature and negative-definite Ricci. [Comparison H.2](comparison-theorems-cut-loci-and-curvature-and-topology.md#theorem-h-2) proves that every connected homogeneous manifold with these properties is simply connected and has global exponential diffeomorphisms. Its proof includes the universal-cover and constant-displacement arguments, so no simple-connectivity assumption is added here. If the odd space is zero, the connected homogeneous space is a point, and the conclusions have their evident zero-dimensional meaning. □

**Proposition D.3 (Duality with the metric fixed).** Inside the complexification of a symmetric Lie algebra, set
\[
\mathfrak g^*=\mathfrak h\oplus i\mathfrak m.
\tag{D.4}
\]
This is a real symmetric Lie algebra. Duality preserves orthogonality and effectiveness, is involutive up to its displayed identification, and exchanges compact and noncompact type for semisimple effective pairs. It preserves the set of infinitesimally invariant complex structures on the odd space. Under \(X\leftrightarrow iX\), with the same positive tangent metric, the dual canonical curvature is the negative of the original.

**Proof.** Complex-bilinear extension of the bracket gives
\[
[A,B]^*=[A,B],\qquad [A,iX]^*=i[A,X],\qquad
[iX,iY]^*=-[X,Y].
\]
These formulas close on (D.4). Jacobi holds because it holds in the complexification. The extension of \(\theta\) preserves this subspace with the indicated positive and negative eigenspaces. Applying the construction again gives
\(\mathfrak h\oplus(-\mathfrak m)=\mathfrak g\), with its original bracket.

The even representation on the whole vector space, identified by \(X\leftrightarrow iX\), is unchanged. Thus its invariant positive form transfers, and so does faithfulness on the odd space. Lemma C.1 proves preservation of effectiveness.

Killing forms are traces of the same complexified adjoint matrices, so their restrictions satisfy
\[
B^*(A,B)=B(A,B),\quad B^*(iX,iY)=-B(X,Y),\quad
B^*(\mathfrak h,i\mathfrak m)=0.
\tag{D.5}
\]
These identities preserve nondegeneracy and exchange the two definiteness conditions. By the Killing criterion in [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) the dual remains semisimple.

The operator \(I^*(iX)=iIX\) has square minus the identity and commutes with the even action exactly when \(I\) does. Compatibility with the transferred positive metric is also identical. This is an infinitesimal statement; an actual global presentation must satisfy the full stabilizer condition (A.3).

Substitute the brackets into (0.1):
\[
R^*(iX,iY)iZ=-[\, -[X,Y],iZ\,]
=i[[X,Y],Z]=-iR(X,Y)Z.
\tag{D.6}
\]
The tangent identification is isometric by the choice of metric, so this reverses every curvature numerator and the Ricci tensor. Semisimple realizations with connected isotropy are supplied by C.4 for either pair. □

## E. The central isotropy element

**Theorem E.1 (A complex structure is an inner derivation).** Let \(G/H\) be a connected symmetric isometry presentation of a Hermitian symmetric space whose Lie-algebra pair is effective and semisimple. Its complex structure is \(G\)-invariant, and its value at the origin is generated by a unique \(Z\in\mathfrak z(\mathfrak h)\):
\[
\operatorname{ad}Z|_{\mathfrak m}=I,\qquad
C_{\mathfrak g}(Z)=\mathfrak h.
\tag{E.1}
\]
Every member of the full stabilizer fixes this element:
\(\operatorname{Ad}(h)Z=Z\) for \(h\in H\).

**Proof.** The even action on the metric odd space is skew and faithful, by C.1. Pulling back the positive trace form on skew operators gives a positive inner product on the even space:
\[
q(A,B)=-\operatorname{tr}_{\mathfrak m}
  \bigl(\operatorname{ad}A|_{\mathfrak m}\operatorname{ad}B|_{\mathfrak m}\bigr).
\]
Cyclicity of trace makes this form invariant under the even adjoint action. Its orthogonal sum with the given odd metric therefore makes the pair orthogonal in the sense of Part C.

By A.2 the complex structure is parallel and every point symmetry is holomorphic. In C.2 the Euclidean ideal of a semisimple algebra must vanish: its abelian odd ideal is zero by semisimplicity, and faithfulness then kills its even part. Thus \(\mathfrak m\) generates \(\mathfrak g\). For \(X\in\mathfrak m\), the action of \(\exp(tX)\) is the transvection along \(\gamma(t)=\exp(tX)H\): [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) and [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) give the same value and parallel-transport differential, and [Affine transformations A.1](affine-transformations-and-the-isometry-group.md#lemma-a-1) gives equality of the global affine maps. That transvection is a product of two point symmetries, hence preserves \(J\). The infinitesimal action of \(\mathfrak m\) therefore preserves \(J\); so do its brackets, by expanding commutators of Lie derivatives. Since these generate \(\mathfrak g\), all its infinitesimal actions preserve \(J\). Their flows, and hence connected \(G\), preserve \(J\), by exponential generation in [Flat connections A.1](flat-connections-and-infinitesimal-holonomy.md#lemma-a-1). This proves the asserted invariance.

The commutator defining curvature, applied to the parallel tensor \(J\), gives
\([R(X,Y),I]=0\) at the origin. The Riemannian curvature symmetries, proved in [Sectional curvature A.1](sectional-curvature-and-space-forms.md#theorem-a-1), and orthogonality of \(I\) imply
\[
\begin{aligned}
b(R(IX,IY)U,V)
 &=b(R(U,V)IX,IY)\\
 &=b(IR(U,V)X,IY)\\
 &=b(R(U,V)X,Y)
 =b(R(X,Y)U,V).
\end{aligned}
\]
Nondegeneracy gives \(R(IX,IY)=R(X,Y)\). The curvature formula (0.1) and faithfulness of the even action from C.1 yield
\[
[IX,IY]=[X,Y].
\]
Replacing \(Y\) by \(IY\) gives
\[
[IX,Y]+[X,IY]=0.
\tag{E.2}
\]

Define \(D|_{\mathfrak h}=0\) and \(D|_{\mathfrak m}=I\). It is a derivation. For two even arguments this is immediate; for one even and one odd it is the infinitesimal invariance of \(I\); for two odd arguments it is (E.2). [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6), whose proof does not assume this geometric assertion, supplies a unique \(Z\in\mathfrak g\) with \(D=\operatorname{ad}Z\).

Since \(D\) commutes with \(\theta\),
\(\operatorname{ad}(\theta Z)=\theta D\theta^{-1}=D\).
The zero centre of the semisimple algebra gives \(\theta Z=Z\), hence \(Z\in\mathfrak h\). Its zero action on \(\mathfrak h\) makes it central there. An element \(A+X\), \(A\in\mathfrak h,X\in\mathfrak m\), commutes with \(Z\) exactly when \(IX=0\), hence exactly when \(X=0\). This proves the centralizer equality.

For \(h\in H\), the adjoint action preserves both eigenspaces and commutes with \(I\) on the odd one, by (A.3). It therefore commutes with \(D\) on all of \(\mathfrak g\). Conjugating the inner derivation gives
\(\operatorname{ad}(\operatorname{Ad}(h)Z)=D\).
Uniqueness of the inner generator proves the last assertion, including disconnected stabilizers. □

**Theorem E.2 (The irreducible central-circle criterion).** Let an effective semisimple orthogonal pair have a nonzero irreducible odd representation. Its connected-isotropy presentation is Hermitian symmetric exactly when
\(\mathfrak z(\mathfrak h)\ne0\).
In this case that centre is one dimensional, and there are exactly two infinitesimally invariant complex structures, \(I\) and \(-I\). They are compatible with every invariant positive tangent metric and transfer to the dual pair. Their adjoint rotations form a circle. Descent through any further disconnected stabilizer is still governed by (A.3).

**Proof.** A Hermitian structure gives the nonzero central element of E.1, so proves necessity. Conversely take \(0\ne Z\in\mathfrak z(\mathfrak h)\), and let \(A=\operatorname{ad}Z|_{\mathfrak m}\). Faithfulness makes \(A\ne0\). It is skew-adjoint for an invariant positive metric and commutes with the isotropy. Its square is self-adjoint and commutes with the isotropy, so its eigenspaces are invariant. Irreducibility and [De Rham decomposition H.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-h-2) give
\[
A^2=-c^2I_{\mathfrak m},\qquad c>0.
\]
Indeed the scalar is nonpositive since
\(\langle A^2X,X\rangle=-|AX|^2\), and scalar zero would force \(A=0\). Thus \(I=A/c\) is an orthogonal complex structure commuting with the even action. For connected isotropy this is full invariance, since exponentials generate that group. Theorem A.3 supplies the Hermitian symmetric metric. Existence of an actual such presentation, rather than only an abstract algebra, is C.4.

If \(W\) is another central even element, its odd action \(A_W\) commutes with \(I\). Both are skew-adjoint, so \(IA_W\) is self-adjoint. It commutes with the isotropy and is therefore a scalar. Multiplication by \(-I\) says \(A_W\) is a scalar multiple of \(I\). Faithfulness then says that \(W\) is a scalar multiple of \(Z\). The centre is one dimensional.

Any infinitesimally invariant complex structure \(J\) can first be made orthogonal using (A.4); connected-isotropy invariance and A.3 then give a Hermitian symmetric structure. Theorem E.1 supplies a central element generating it. The one-dimensional centre makes \(J=tI\), and \(J^2=-I_{\mathfrak m}\) gives \(t=\pm1\). Finally D.1 makes every invariant positive metric on this irreducible odd space a positive scalar multiple of any one such metric, so these two structures are orthogonal for all of them.

Normalize the central generator so that its odd action is \(I\). Its adjoint exponential is the identity on \(\mathfrak h\) and
\[
e^{tI}=(\cos t)I_{\mathfrak m}+(\sin t)I
\]
on \(\mathfrak m\), by separating the even and odd terms of the exponential series. The circle parameter proved in [Connections E.1](connections-and-parallel-transport.md#lemma-e-1) shows that its least positive period is \(2\pi\), since \(\mathfrak m\ne0\) and \(I\) is not a real scalar. Its image is a compact circle of automorphisms. Proposition D.3 transfers the two complex structures to the dual. A further global quotient preserves one of them precisely when its full stabilizer commutes with it, by A.3. □

## F. Two compact-group foundations

**Lemma F.1 (The universal covering group).** The connected universal cover of a connected Lie group has a unique Lie-group structure for which its based covering map is a homomorphism. The map identifies the Lie algebras and has discrete central kernel.

**Proof.** [Flat connections D.2](flat-connections-and-infinitesimal-holonomy.md#theorem-d-2) constructs the connected smooth universal cover \(p:\widetilde G\to G\). Choose \(\widetilde e\) over the identity. The product \(\widetilde G\times\widetilde G\) is simply connected: contract each coordinate of a loop separately. The map
\((x,y)\mapsto p(x)p(y)\) therefore has a unique based lift
\(\widetilde m:\widetilde G\times\widetilde G\to\widetilde G\), by the lifting theorem [Flat connections D.3](flat-connections-and-infinitesimal-holonomy.md#theorem-d-3). Lift inversion in the same way. These lifts are smooth, since in every evenly covered neighbourhood they are compositions with a smooth local inverse of \(p\).

Both associativity maps on \(\widetilde G^3\) lift the same map into \(G\), and agree at the triple identity. Uniqueness of lifts on a connected domain makes them equal. The maps \(x\mapsto\widetilde m(\widetilde e,x)\) and \(x\mapsto\widetilde m(x,\widetilde e)\) likewise equal the identity. Multiplying \(x\) by its lifted inverse maps into the discrete fibre over the identity; it is constant because \(\widetilde G\) is connected, and its value at \(\widetilde e\) is \(\widetilde e\). These are the group axioms. Any other covering-group operations would be the same based lifts, proving uniqueness.

The local diffeomorphism \(p\) preserves brackets by differentiation, and its differential is an isomorphism. Its kernel is discrete by the covering charts. For a kernel element \(\gamma\), the continuous map \(x\mapsto x\gamma x^{-1}\) takes the connected group into that discrete kernel. It is constant and equals \(\gamma\) at the identity, proving centrality. □

**Theorem F.2 (Negative Killing form and compact covers).** If a connected Lie group has negative-definite Killing form, then the group and its simply connected covering group are compact. Their exponential maps are surjective.

**Proof.** Use the positive form \(-B\) at the identity. Killing invariance from [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) makes every adjoint infinitesimal operator skew for this form. Exponentiation, followed by generation of the connected group by exponentials, proves invariance under the full adjoint action. [Homogeneous spaces C.3](homogeneous-spaces-and-invariant-connections.md#example-c-3) therefore gives a bi-invariant metric with complete Levi-Civita connection
\[
\nabla_{X^L}Y^L=\tfrac12[X,Y]^L,\qquad
R(X,Y)Z=-\tfrac14[[X,Y],Z].
\]
Tracing \(Z\mapsto R(Z,X)Y\) gives
\[
\operatorname{Ric}(X,Y)=-\tfrac14 B(X,Y)=\tfrac14 g(X,Y).
\tag{F.1}
\]
The trace identity follows directly from
\([[Z,X],Y]=\operatorname{ad}Y\,\operatorname{ad}X\,Z\).

If the dimension is zero the connected group is a point. Dimension one is impossible here: a one-dimensional Lie algebra is abelian and has zero Killing form. In dimension \(n\geq2\), [Comparison D.1](comparison-theorems-cut-loci-and-curvature-and-topology.md#theorem-d-1) applies to (F.1) with \(k=1/(4(n-1))\), proving compactness. Lemma F.1 constructs the universal covering group with the same Lie algebra. The identical metric and trace argument, or the covering part of [Comparison D.1](comparison-theorems-cut-loci-and-curvature-and-topology.md#theorem-d-1), proves it compact too.

For either group, Hopf–Rinow B.2 joins the identity to any prescribed point by a minimizing geodesic. The geodesics through the identity for the displayed connection are \(t\mapsto\exp(tX)\), by [Homogeneous spaces C.3](homogeneous-spaces-and-invariant-connections.md#example-c-3). Evaluating at time one proves surjectivity of the exponential. □

**Theorem F.3 (Compact abelian groups and a dense cyclic subgroup).** A connected compact abelian Lie group is a torus
\[
\mathbb R^r/\Lambda
\]
for a full lattice \(\Lambda\). It contains an element whose integer powers are dense. A closed subgroup of a finite-dimensional Lie group is an embedded Lie subgroup, as proved in Invariant connections A.1.

**Proof.** The Lie algebra of an abelian group is abelian, since its invariant flows commute. Its exponential is a homomorphism from the additive Lie algebra: the two curves \(\exp(t(X+Y))\) and \(\exp(tX)\exp(tY)\) have the same initial value and invariant velocity, so agree by [Local tools 2.3](local-tools-for-bundles-and-transport.md#2-differential-equations-and-their-parameters). The differential of the exponential at zero is the identity, obtained by differentiating \(t\mapsto\exp(tX)\). [Local tools 1.2](local-tools-for-bundles-and-transport.md#1-contraction-and-local-inversion) makes it a local diffeomorphism. Its image is an open subgroup, hence all of the connected group. Its kernel \(\Lambda\) is discrete. The induced bijection \(\mathbb R^r/\Lambda\to G\) is a local diffeomorphism in quotient charts and hence a diffeomorphism and group isomorphism. [Sectional curvature G.3](sectional-curvature-and-space-forms.md#theorem-g-3) gives an integral basis of the discrete group and proves that compactness of this quotient is equivalent to full rank. A linear change of basis reduces to \(\Lambda=\mathbb Z^r\).

Choose real numbers \(\alpha_1,\ldots,\alpha_r\) such that
\(1,\alpha_1,\ldots,\alpha_r\) are linearly independent over \(\mathbb Q\).
Such a finite choice is possible recursively: the rational span of finitely many real numbers is countable, whereas \(\mathbb R\) is uncountable by the binary-sequence diagonal argument given in [Symmetric spaces C.4](symmetric-spaces.md#theorem-c-4). Let \(a\) be the class of \(\alpha=(\alpha_1,\ldots,\alpha_r)\) in the torus, and \(C\) the closure of its integer powers. Continuity of multiplication and inversion makes \(C\) a subgroup; it is closed, hence a compact embedded Lie subgroup by Invariant connections A.1.

Its identity component \(C^0\) is compact, connected and abelian, so the first paragraph applies to it. Write \(W\subseteq\mathbb R^r\) for its Lie algebra. The exponential of \(C^0\) is the restriction of the torus exponential, by uniqueness of the invariant ODE, so
\[
C^0=\exp(W),\qquad
C^0\simeq W/(W\cap\mathbb Z^r).
\]
Its compactness makes \(W\cap\mathbb Z^r\) span \(W\) by [Sectional curvature G.3](sectional-curvature-and-space-forms.md#theorem-g-3). If \(W\ne\mathbb R^r\), choose a nonzero integer vector \(m\) annihilating \(W\): an integral basis of this lattice gives a matrix of rank less than \(r\), whose homogeneous equations have a nonzero rational solution by Gaussian elimination; clearing denominators gives \(m\).

Then the torus character
\(\chi_m([x])=e^{2\pi i m\cdot x}\)
is one on \(C^0\). Components of the Lie group \(C\) are open, since an identity coordinate ball is connected and translates give neighbourhoods in each component. Compactness gives finitely many components, say \(N\). In the finite group \(C/C^0\), the order of each element is at most \(N\), by repetition among its first \(N+1\) powers, so its \(N!\)-th power is the identity. It follows that \(\chi_{N!m}\) is one on \(C\). In particular,
\[
N!\,m\cdot\alpha\in\mathbb Z,
\]
contradicting the chosen rational independence. Therefore \(W=\mathbb R^r\). The subgroup \(C^0\), and hence \(C\), is the whole torus. This proves density. For \(r=0\) the torus is a point and its identity has the required property. □

## G. Exercises with solutions

**Exercise G.1 (Sphere, hyperboloid and scale).** Give the compact and noncompact duality for the round sphere and real hyperbolic space, including the curvature normalization and the one-dimensional case.

**Solution.** For \(n\geq2\), the sphere presentation has algebra
\(\mathfrak{so}(n+1)=\mathfrak{so}(n)\oplus\mathfrak m\), with
\[
X(x)=\begin{pmatrix}0&-x^T\\x&0\end{pmatrix},
\qquad x\in\mathbb R^n.
\]
Its odd-odd bracket has bottom-right block
\(-xy^T+yx^T\). In the hyperboloid presentation for the identity component of the group preserving \(\operatorname{diag}(-1,I_n)\), the odd matrices are
\[
Y(x)=\begin{pmatrix}0&x^T\\x&0\end{pmatrix},
\]
and their odd-odd bracket has bottom-right block
\(xy^T-yx^T\). The even brackets and even actions on \(x\) agree. Thus these are exactly the dual brackets of D.3.

The sphere presentation is proved in [Symmetric spaces E.1](symmetric-spaces.md#example-e-1). For the future unit hyperboloid, the Lorentz boost
\(\left(\begin{smallmatrix}\cosh t&\sinh t\\\sinh t&\cosh t\end{smallmatrix}\right)\)
on the plane spanned by the time axis and a spatial unit vector \(u\), extended by the identity on its orthogonal spatial complement, is a path from the identity preserving the form. It carries the base point to \((\cosh t,\sinh t\,u)\), and these points exhaust that sheet. Its stabilizer consists exactly of \(\operatorname{diag}(1,A)\), \(A\in SO(n)\); these rotations lie in the identity component by the plane-rotation argument in [Affine transformations G.1](affine-transformations-and-the-isometry-group.md#theorem-g-1). This proves the stated hyperboloid presentation as well.

With tangent inner product \(x\cdot y\), substitution in (0.1) gives
\[
R_c(x,y)z=(y\cdot z)x-(x\cdot z)y,\qquad
R_n(x,y)z=-R_c(x,y)z.
\]
For orthonormal \(x,y\) the sectional curvatures are respectively \(1\) and \(-1\). Multiplying the metric by \(r^2>0\) leaves its Levi-Civita connection unchanged, by the Koszul formula: both sides acquire the same factor. The curvature numerator acquires \(r^2\), and the two-plane squared-area denominator acquires \(r^4\), giving \(r^{-2}\) and \(-r^{-2}\). [Sectional curvature C.1–C.3](sectional-curvature-and-space-forms.md#theorem-c-1) proves that these are the radius-\(r\) round sphere and hyperboloid models, with their completeness and simple connectivity for \(n\geq2\).

When \(n=1\), the even algebra is zero and both displayed odd algebras are one-dimensional and abelian. Their curvature tensors vanish; there is no tangent two-plane. The global spaces are a circle and a line, and the algebraic duality leaves the same flat one-dimensional algebra. The compact/noncompact semisimple terminology of Part C therefore does not classify these one-dimensional models as a nonzero pair of opposite curvature types. □

**Exercise G.2 (Which hypothesis removes invisible isotropy?).** Prove the orthogonal decomposition, explain the Euclidean rotations, and determine what changes if effectiveness is replaced by
\(\mathfrak h\cap Z(\mathfrak g)=0\).

**Solution.** For an effective pair, C.1 gives negative-definite even Killing form. Form the self-adjoint Killing operator on \(\mathfrak m\). The proof of C.2 shows that distinct eigenspaces commute, that the zero eigenspace is abelian and commutes with all odd vectors, and that every nonzero eigenspace generates a semisimple symmetric ideal. The signs of its eigenvalues distinguish compact and noncompact type. Decomposing each nonzero eigenspace into orthogonal invariant irreducible subspaces and repeating that bracket argument gives the irreducible blocks.

The even orthogonal complement of the brackets of all nonzero eigenspaces may still act on the zero eigenspace. It acts there faithfully and by skew operators, giving
\(\mathfrak h_0\ltimes\mathfrak m_0\), not just a translation algebra. For example, take
\[
\mathfrak g=\mathfrak{so}(n)\ltimes\mathbb R^n,\qquad
\theta(A,x)=(A,-x).
\]
For \(n\geq2\) the standard skew action is faithful: if \(Ax=0\) for every \(x\), the matrix is zero. The positive form
\(-\operatorname{tr}(AB)+x\cdot y\) is invariant under the even action, by cyclicity of trace and skew-adjointness. Odd brackets vanish, so this is an effective Euclidean pair retaining nonzero rotations. If instead the odd space is required to generate the whole algebra, C.2 proves that those rotations disappear.

Under the weaker central-intersection hypothesis, C.3 first splits off the kernel of the even action. It is a compact semisimple ideal lying entirely in \(\mathfrak h\); the remaining pair is effective, so the preceding construction applies. Without either hypothesis one can add a nonzero central even abelian summand acting trivially on everything. Its Killing form is zero, so it cannot be assigned to a compact semisimple ideal, and its invisible isotropy is not faithful Euclidean isotropy. This verifies why the hypothesis is needed for the stated decomposition. □

## H. Maximal tori and connected centralizers

A torus in a Lie group is a compact connected abelian Lie subgroup. Part F identifies such a group with a real vector space modulo a full lattice. A maximal torus means one maximal by inclusion.

**Lemma H.1 (Exponential surjectivity and containing tori).** In a connected compact Lie group \(K\), every element is an exponential and belongs to a maximal torus. Every torus in \(K\) is contained in a maximal torus.

**Proof.** [Symmetric spaces D.1](symmetric-spaces.md#lemma-d-1) constructs a positive \(\operatorname{Ad}(K)\)-invariant inner product \(Q\) by integration. Its left-translated metric is bi-invariant, and [Homogeneous spaces C.3](homogeneous-spaces-and-invariant-connections.md#example-c-3) proves that its Levi-Civita connection is
\[
\nabla_{X^L}Y^L=\tfrac12[X,Y]^L.
\]
That proof also shows that this connection is geodesically complete and that its geodesics from the identity are \(t\mapsto\exp(tX)\). Hopf–Rinow B.2 joins the identity to each \(k\in K\) by one of these geodesics. Thus \(k=\exp X\) for some \(X\in\mathfrak k\).

The closure of \(\{\exp(tX):t\in\mathbb R\}\) is a subgroup: continuity of multiplication and inversion shows that products and inverses of limits remain in the closure. It is abelian for the same reason. It is connected, since the closure of a connected set is connected, and compact as a closed subset of \(K\). To justify the connectedness fact, a separation of that closure into two nonempty relatively open sets would separate the original connected dense subset; density ensures that both sets meet it. The closed subgroup theorem, Invariant connections A.1, makes the closure an embedded Lie subgroup. It is therefore a torus containing \(k\).

Starting with any torus \(S\), choose among the tori containing it one of largest dimension. Such a choice uses only a maximum of a nonempty set of integers between zero and \(\dim K\). A proper inclusion of tori increases dimension: if two such tori have equal dimension, their Lie algebras agree, so uniqueness of the connected subgroup in [Flat connections A.1](flat-connections-and-infinitesimal-holonomy.md#lemma-a-1) makes them equal. The chosen torus is thus maximal. Apply this also to the torus constructed from \(X\). The zero-dimensional connected group is the trivial group, for which all conclusions hold. □

**Theorem H.2 (Conjugacy of maximal tori and central elements).** All maximal tori of a connected compact Lie group are conjugate. Every central element belongs to every maximal torus.

**Proof.** Let \(T,S\subseteq K\) be maximal tori. By F.3 choose elements \(a\in T\), \(b\in S\) whose integer powers are dense in their respective tori. The exponential descriptions in F.3 give \(X\in\mathfrak t\), \(Y\in\mathfrak s\) with
\[
a=\exp X,\qquad b=\exp Y.
\]
Use the invariant positive form \(Q\) from [Symmetric spaces D.1](symmetric-spaces.md#lemma-d-1). The smooth real function
\[
f(k)=Q(\operatorname{Ad}(k)X,Y)
\]
attains a maximum at some \(k_0\), by compactness. Write \(U=\operatorname{Ad}(k_0)X\). Differentiating along \(\exp(tA)k_0\), for arbitrary \(A\in\mathfrak k\), gives
\[
0=Q([A,U],Y)=Q(A,[U,Y]).
\tag{H.1}
\]
The derivative of the adjoint action is proved in [Curvature and holonomy A.3](curvature-and-holonomy-groups.md#lemma-a-3). The second equality follows by differentiating invariance of \(Q\). Its nondegeneracy gives \([U,Y]=0\).

Commuting Lie algebra elements have commuting exponential curves. Indeed the adjoint curve satisfies
\[
\frac{d}{dt}\operatorname{Ad}(\exp(tU))Y
 =[U,\operatorname{Ad}(\exp(tU))Y];
\]
the constant value \(Y\) solves this equation, so uniqueness in [Local tools 2.3](local-tools-for-bundles-and-transport.md#2-differential-equations-and-their-parameters) proves it is fixed. Naturality of the exponential, again from the invariant ODE, then proves the assertion. In particular \(k_0ak_0^{-1}\) commutes with \(b\).

Taking integer powers and then limits proves that every element of \(k_0Tk_0^{-1}\) commutes with every element of \(S\). Their product is an abelian subgroup. It is connected and compact, being the continuous image of the product of the two compact connected tori. Thus it is closed in the Hausdorff group and an embedded torus by Invariant connections A.1. Maximality forces
\[
S=k_0Tk_0^{-1}.
\]

Let \(z\in Z(K)\), and fix a maximal torus \(T\). By H.1 some maximal torus \(S\) contains \(z\). Choose \(k\) with \(kSk^{-1}=T\). Centrality gives \(kzk^{-1}=z\), so \(z\in T\). This proves the second assertion without assuming that maximal tori are already self-centralizing. □

**Theorem H.3 (Connected centralizers).** If \(S\) is a torus in a connected compact Lie group \(K\), then
\[
C_K(S)=\{k\in K:ks=sk\text{ for every }s\in S\}
\tag{H.2}
\]
is connected. More precisely it is the union of the maximal tori of \(K\) containing \(S\). For every \(Z\in\mathfrak k\), the group
\[
C_K(Z)=\{k\in K:\operatorname{Ad}(k)Z=Z\}
\tag{H.3}
\]
is connected, with Lie algebra \(\{X:[X,Z]=0\}\).

**Proof.** Fix \(c\in C_K(S)\). The centralizer of the single group element \(c\) is a closed subgroup, hence an embedded compact Lie subgroup by Invariant connections A.1. Let
\[
L=C_K(c)^0
\]
be its identity component. This component is closed: the closure of a connected subset is connected, so maximality of a component contains its closure. Thus \(L\) is a compact connected Lie group.

First \(c\) itself belongs to \(L\). By H.1 write \(c=\exp X\). The path \(\exp(tX)\), \(0\leq t\leq1\), commutes with \(c\) and joins the identity to \(c\). Also \(S\subseteq L\), because \(S\) is connected, contains the identity and commutes with \(c\).

By H.1 applied inside \(L\), choose a maximal torus \(T_L\subseteq L\) containing \(S\). The element \(c\) is central in \(L\) by its definition. Theorem H.2, also applied inside \(L\), therefore puts \(c\) in \(T_L\). Enlarge \(T_L\) to a maximal torus \(T\) of \(K\) using H.1. This torus contains both \(S\) and \(c\), and, being abelian, lies in \(C_K(S)\).

Conversely every torus containing \(S\) lies in its centralizer. The claimed union follows. Each torus in the union is connected and contains the identity; their union is connected. Indeed a separation of the union would separate one of these tori from the common identity, which is impossible. This proves (H.2).

For (H.3), let \(S_Z\) be the closure of \(\exp(\mathbb R Z)\). It is a torus by the closure argument in H.1. Exponential naturality shows that an element fixing \(Z\) centralizes this one-parameter subgroup and its closure. Conversely an element centralizing the closure fixes \(Z\) by differentiating at \(t=0\). Thus
\[
C_K(Z)=C_K(S_Z),
\]
which is connected. Differentiating \(\operatorname{Ad}(\exp(tX))Z=Z\) proves the necessary Lie algebra condition. If \([X,Z]=0\), the adjoint ODE has constant solution \(Z\), so \(\exp(tX)\) lies in the centralizer for every \(t\); this proves sufficiency. The Lie algebra assertion follows from the closed subgroup characterization in Invariant connections A.1. □

## I. The global compact Hermitian conclusion

**Lemma I.1 (A simply connected group modulo connected isotropy).** Let \(K\) be a connected simply connected Lie group and \(L\subseteq K\) a closed connected subgroup. Then \(K/L\) is simply connected.

**Proof.** [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) constructs the smooth quotient, its local sections and its principal-bundle product charts. It is connected as an image of \(K\), and it is path connected: a connected manifold is path connected by the coordinate-ball argument in A.2.

Let \(\alpha:[0,1]\to K/L\) be a continuous loop based at \(o=eL\). It has a continuous lift \(u:[0,1]\to K\) with \(u(0)=e\). Here only path lifting, not a covering-space assumption, is needed. To construct it, subdivide the interval into finitely many subintervals on each of which the image of \(\alpha\) lies in a domain of a local section. Such a subdivision exists by compactness: a finite open cover of the interval has a positive Lebesgue number. For completeness, if no positive number worked, intervals of lengths tending to zero failing to lie in any cover member would have centre points with a convergent subsequence; a neighbourhood of the limiting point lying in one cover member would eventually contain those whole intervals, a contradiction. On one subinterval, a section \(s\) gives a lift \(s(\alpha(t))h\), with constant \(h\in L\) uniquely chosen to equal the already assigned lift at its starting endpoint. This constructs the lift successively and continuously.

Its endpoint \(u(1)\) lies in \(L\). Since \(L\) is a connected manifold, join \(u(1)\) to \(e\) by a path \(v\) in \(L\). Concatenating \(u\) and \(v\) gives a loop in simply connected \(K\), and hence a based contraction. Project that contraction to \(K/L\). Its initial loop is \(\alpha\) followed by a constant segment at \(o\). Removing that segment by continuous reparametrization is a based homotopy to \(\alpha\). Thus every based loop contracts. □

**Theorem I.2 (Compact Hermitian symmetric spaces are simply connected).** A connected Hermitian symmetric space of compact type is compact and simply connected. No irreducibility assumption or connectedness assumption on its stabilizer is required.

**Proof.** Choose its connected effective symmetric isometry presentation \(G/H\), as in [Symmetric spaces B.3](symmetric-spaces.md#theorem-b-3). Compact type means that the Killing form of this semisimple pair is negative definite. Theorem F.2 therefore makes \(G\), and also its simply connected covering group
\[
p:\widetilde G\longrightarrow G,
\]
compact. In particular \(G/H\) is compact as a continuous image of \(G\). Lemma F.1 identifies the two Lie algebras by \(dp_e\).

Theorem E.1 supplies \(Z\in\mathfrak z(\mathfrak h)\) with
\[
\operatorname{ad}Z|_{\mathfrak m}=I,\qquad
C_{\mathfrak g}(Z)=\mathfrak h,\qquad
H\subseteq C_G(Z).
\tag{I.1}
\]
The last inclusion holds for the full stabilizer, including all components. By H.3 the group \(C_G(Z)\) is connected and has Lie algebra \(\mathfrak h\). Its connected subgroup with this algebra is unique by [Flat connections A.1](flat-connections-and-infinitesimal-holonomy.md#lemma-a-1). Consequently \(H^0=C_G(Z)\). Together with (I.1) this gives
\[
H=C_G(Z),
\tag{I.2}
\]
so connectedness of the stabilizer has been proved.

Let \(\widetilde Z=(dp_e)^{-1}Z\). Differentiation of
\[
p(kgk^{-1})=p(k)p(g)p(k)^{-1}
\]
gives equivariance of \(dp_e\) under the two adjoint actions. Since this differential is injective, (I.2) implies the exact equality
\[
p^{-1}(H)=C_{\widetilde G}(\widetilde Z).
\tag{I.3}
\]
The group on the right is connected by H.3, applied to the compact connected group \(\widetilde G\). It is closed, being a stabilizer under a continuous linear action.

The map induced by \(p\) identifies
\[
\widetilde G/p^{-1}(H)\ \longrightarrow\ G/H
\tag{I.4}
\]
diffeomorphically. It is onto since \(p\) is onto, and equality of image cosets is exactly membership in \(p^{-1}(H)\), proving injectivity. In the quotient charts of [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) its differential is the induced isomorphism of tangent quotient spaces, because \(dp_e\) is an isomorphism; translate this at the identity coset to every point. Thus it is a local diffeomorphism by [Local tools 1.2](local-tools-for-bundles-and-transport.md#1-contraction-and-local-inversion). Its bijectivity makes the local smooth inverses agree, proving the assertion.

Apply I.1 to the simply connected \(\widetilde G\) and the connected closed subgroup in (I.3). The quotient in (I.4), and hence the given space, is simply connected. The proof uses the single central generator of the full complex structure from E.1; it does not require a decomposition into irreducible spaces. For a zero-dimensional connected space the result says that a point is compact and simply connected. □

## J. A covering-space exercise

**Exercise J.1 (Prove connectedness before using it).** Prove that every connected Hermitian symmetric space of compact type is simply connected, allowing a disconnected stabilizer at the outset. Identify where compactness of the universal covering group is needed. Explain why mere compactness of a Hermitian symmetric space would not suffice.

**Solution.** Begin with its effective connected presentation \(G/H\). Negative-definite Killing form gives the compact group and compact universal covering group by F.2; this is proved with the bi-invariant metric, its positive Ricci trace and Myers' theorem, not by assuming that covers of compact spaces are compact.

The central element \(Z\) in E.1 is fixed by every element of \(H\), and its Lie algebra centralizer is exactly \(\mathfrak h\). Theorem H.3 makes \(C_G(Z)\) connected, so equality of Lie algebras and the inclusions \(H^0\subseteq H\subseteq C_G(Z)\) force \(H=C_G(Z)\). On the universal covering group, adjoint equivariance gives \(p^{-1}(H)=C_{\widetilde G}(\widetilde Z)\). Compactness of \(\widetilde G\) is used to apply H.3 again, proving that this entire inverse image is connected. Lemma I.1 then contracts loops in its quotient by lifting a loop, closing the lift inside that subgroup, and projecting a contraction upstairs. This is the argument of I.2 with every connectedness step justified.

For the last question, let
\[
M=\mathbb C/(\mathbb Z+i\mathbb Z).
\]
Translations preserve the Euclidean Hermitian metric and the complex coordinate structure, so they descend through the lattice quotient. Reflection about any \(p\in M\), namely \(z\mapsto2p-z\) modulo the lattice, is a well-defined holomorphic isometry fixing \(p\) with derivative \(-I\). The image of the closed unit square is all of \(M\), proving compactness.

The path \(t\mapsto[t]\), \(0\leq t\leq1\), is a loop whose lift to the plane starts at zero and ends at one. The quotient by the lattice is a covering: discs of radius less than \(1/3\) have disjoint lattice translates, giving its evenly covered charts. If the loop contracted, covering homotopy lifting, proved in [Flat connections D.1](flat-connections-and-infinitesimal-holonomy.md#lemma-d-1), would keep the endpoint of its based lift fixed in the discrete fibre during that contraction. A constant loop has endpoint zero, a contradiction. Hence \(M\) is not simply connected. Its translation algebra is abelian, so its Killing form is zero; this compact example has Euclidean rather than compact semisimple type. □

## K. Matrix square roots and unitary paths

For a complex matrix, \(X^*=\bar X^T\). A Hermitian matrix \(P=P^*\) is positive definite when \(v^*Pv>0\) for every nonzero complex column \(v\). On rectangular matrices put \(\|X\|_{\mathrm{HS}}^2=\operatorname{tr}(X^*X)=\sum_{a,b}|X_{ab}|^2\).

**Lemma K.1 (The matrix tools used below).** A Hermitian matrix has an orthonormal complex eigenbasis and real eigenvalues. A positive-definite matrix has a unique positive-definite square root, which depends smoothly on the matrix. The groups \(U(n)\) and \(SU(n)\) are connected. For any rectangular \(Z\), the positive nonzero eigenvalues of \(Z^*Z\) and \(ZZ^*\), with multiplicities, agree. If \(I-Z^*Z>0\), then \(I-ZZ^*>0\).

**Proof.** Regard a Hermitian operator \(P\) as a self-adjoint real operator for the real inner product \(\operatorname{Re}(v^*w)\). The real spectral theorem, proved in [De Rham decomposition H.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-h-2), gives an orthogonal decomposition into real eigenspaces with real eigenvalues. Each eigenspace is complex linear since \(P\) commutes with multiplication by \(i\). Inside each eigenspace choose a complex orthonormal basis by the following finite induction: normalize a nonzero vector, subtract its complex scalar projection from all remaining vectors, and repeat in its complex orthogonal complement. Real-orthogonal distinct complex eigenspaces are complex orthogonal, since their real orthogonality also holds after multiplying one vector by \(i\). This proves the first assertion.

For positive \(P\), all its eigenvalues \(\lambda\) are positive. Define \(P^{1/2}\) to be multiplication by \(\sqrt{\lambda}\) on each eigenspace. If \(R>0\) and \(R^2=P\), then \(R\) commutes with \(P\); on a \(P\)-eigenspace the eigenvalues of \(R\) are positive and have square \(\lambda\), so \(R=\sqrt{\lambda}I\). This proves uniqueness. To prove smoothness without choosing smooth eigenvectors, consider \(F(R)=R^2\) on Hermitian matrices. At \(R>0\),
\[
dF_R(H)=RH+HR.
\tag{K.1}
\]
In an orthonormal eigenbasis of \(R\), this multiplies the \(a,b\) entry of \(H\) by \(r_a+r_b>0\); it is an invertible real linear map of the Hermitian matrices. Positivity is open: \(v^*Rv\geq m|v|^2\) for the least positive eigenvalue \(m\), and a perturbation of operator norm less than \(m\) preserves positivity. [Local tools 1.2](local-tools-for-bundles-and-transport.md#1-contraction-and-local-inversion) gives smooth local inverse maps for \(F\). Uniqueness makes them agree, proving global smoothness on the positive cone. The same proof on real symmetric matrices gives real symmetric square roots.

If \(Q\in U(n)\), write \(Q=A+iB\), where
\[
A=\tfrac12(Q+Q^*),\qquad B=\tfrac1{2i}(Q-Q^*).
\]
These are commuting Hermitian operators. Diagonalize \(A\), then diagonalize the restriction of \(B\) to each \(A\)-eigenspace; commutation makes those restrictions legitimate. Thus
\(Q=V\operatorname{diag}(e^{it_1},\ldots,e^{it_n})V^*\) with \(V\) unitary and real \(t_j\). The unit-circle parameterization is [Connections E.1](connections-and-parallel-transport.md#lemma-e-1). Replacing each \(t_j\) by \(st_j\), \(0\leq s\leq1\), gives a path from \(I\) to \(Q\). If \(\det Q=1\), their sum is \(2\pi k\); replace one angle by \(t_j-2\pi k\), leaving \(Q\) unchanged and making the sum zero. The resulting path lies in \(SU(n)\).

Finally, for \(\lambda>0\), the map \(v\mapsto Zv\) identifies the \(\lambda\)-eigenspace of \(Z^*Z\) with that of \(ZZ^*\). Its inverse there is \(w\mapsto\lambda^{-1}Z^*w\). Both operators are positive semidefinite since \(v^*Z^*Zv=|Zv|^2\). Their nonzero eigenvalues and multiplicities therefore agree, and any remaining eigenvalues are zero. In particular, if all eigenvalues of \(Z^*Z\) are less than one, the same holds for \(ZZ^*\). This proves the last assertion. These eigenspace identities also give
\[
Zf(Z^*Z)=f(ZZ^*)Z
\tag{K.2}
\]
for any function \(f\) defined on the two finite spectra: on the zero eigenspaces both sides vanish, and on each nonzero eigenspace they agree. We will use this for positive square roots and inverse square roots. □

## L. Complex Grassmannians, matrix balls and their curvature

Fix positive integers \(p,q\), and write \(\mathbb C^{p+q}=\mathbb C^p\oplus\mathbb C^q\). Matrices \(Z\) below have size \(q\times p\). A graph means the \(p\)-plane
\[
E_Z=\{(v,Zv):v\in\mathbb C^p\}.
\tag{L.1}
\]

**Theorem L.1 (The compact Grassmannian in graph coordinates).** The space \(\mathrm{Gr}_p(\mathbb C^{p+q})\) of complex \(p\)-planes is a compact connected complex manifold
\[
U(p+q)/(U(p)\times U(q)).
\tag{L.2}
\]
It has a complete Hermitian symmetric metric whose expression in a graph chart is
\[
\begin{aligned}
P_+(Z)&=I_p+Z^*Z,\qquad Q_+(Z)=I_q+ZZ^*,\\
h_Z^+(U,V)&=\operatorname{tr}\bigl(P_+(Z)^{-1}U^*Q_+(Z)^{-1}V\bigr),\\
g_Z^+(U,V)&=\operatorname{Re}h_Z^+(U,V).
\end{aligned}
\tag{L.3}
\]
Its symmetry at \(E=\mathbb C^p\oplus0\) is \(Z\mapsto-Z\), and its symmetry at an arbitrary plane is the action on planes of the unitary reflection that is \(+I\) on that plane and \(-I\) on its orthogonal complement.

**Proof.** Extend an orthonormal basis of a \(p\)-plane to one of the whole space, using the finite orthogonal-complement induction in K.1. This proves transitivity of \(U(p+q)\); its stabilizer is exactly the block-diagonal subgroup \(U(p)\times U(q)\). The unitary group is closed and bounded in the real matrix space, hence compact by [Local tools 0.1](local-tools-for-bundles-and-transport.md#0-analytic-and-linear-foundations), and connected by K.1. Its polynomial equations make it a closed Lie subgroup by Invariant connections A.1. [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) supplies the smooth quotient in (L.2). It is compact and connected.

Planes whose projection onto the first summand is an isomorphism have the unique graph description (L.1). These are actual smooth quotient coordinates. For example a smooth unitary representative carrying \(E\) to \(E_Z\) is
\[
k_Z=
\begin{pmatrix}
P_+^{-1/2}&-Z^*Q_+^{-1/2}\\
ZP_+^{-1/2}&Q_+^{-1/2}
\end{pmatrix}.
\tag{L.4}
\]
The columns have orthonormal blocks: their diagonal inner products are \(I_p,I_q\), and their cross inner product is zero. Thus \(k_Z^*k_Z=I\). Smoothness follows from K.1. Conversely, if a representative has first column blocks \(a,c\) with \(a\) invertible, its graph coordinate is \(ca^{-1}\), independent of the representative. Invertibility is open, so this chart and its inverse are smooth in the quotient charts. Every plane has such a chart after a unitary change of splitting.

For a fixed block matrix \(k=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), the coordinate action on its graph domain is
\[
W=(c+dZ)(a+bZ)^{-1}.
\tag{L.5}
\]
It is holomorphic: the entries of an inverse matrix are polynomial cofactors divided by its nonzero determinant. These transition maps give a complex atlas of dimension \(pq\), and the full unitary action is holomorphic.

Write \(L=a+bZ\) and \(T=d-Wb\). Differentiating (L.5) gives
\[
dW(U)=TUL^{-1}.
\tag{L.6}
\]
Preservation of the column inner product of \(\binom{I}{Z}\) gives
\[
P_+(W)=L^{-*}P_+(Z)L^{-1}.
\tag{L.7}
\]
The other required identity is
\[
Q_+(W)=TQ_+(Z)T^*.
\tag{L.8}
\]
To verify it, note \(TZ=Wa-c\), then expand
\[
\begin{aligned}
T(I+ZZ^*)T^*
&=(d-Wb)(d^*-b^*W^*)+(Wa-c)(a^*W^*-c^*)\\
&=dd^*+cc^*+W(bb^*+aa^*)W^*\\
&\quad-(db^*+ca^*)W^*-W(bd^*+ac^*)\\
&=I+WW^*.
\end{aligned}
\]
Here the block identities in the last step are \(kk^*=I\). In particular \(T\) is invertible. Substitute (L.6)–(L.8) into (L.3); adjacent inverse factors cancel, and cyclicity of trace leaves \(h_Z^+(U,V)\). Thus the formulas agree on overlaps and define an invariant smooth Hermitian form. Positivity is explicit:
\[
h_Z^+(U,U)=\|Q_+^{-1/2}UP_+^{-1/2}\|_{\mathrm{HS}}^2>0
\quad(U\ne0).
\tag{L.9}
\]

The unitary matrix \(\operatorname{diag}(I_p,-I_q)\) acts as \(Z\mapsto-Z\), hence fixes \(E\) and has derivative \(-I\) there. Its conjugate at another plane is precisely the stated reflection. All these maps are holomorphic isometries. Theorem A.2 makes the metric Kähler; [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves completeness from these global point symmetries. This proves every metric and global assertion. □

**Theorem L.2 (The noncompact matrix ball).** Put
\[
\mathcal D_{q,p}=\{Z\in M_{q,p}(\mathbb C):P_-(Z)=I_p-Z^*Z>0\},
\qquad Q_-(Z)=I_q-ZZ^*.
\tag{L.10}
\]
This is a bounded convex complex domain. The group
\[
U(p,q)=\{g:g^*\operatorname{diag}(I_p,-I_q)g
                  =\operatorname{diag}(I_p,-I_q)\}
\tag{L.11}
\]
acts transitively and holomorphically by (L.5), now defined for every \(Z\in\mathcal D_{q,p}\). The stabilizer of zero is \(U(p)\times U(q)\). The metric
\[
h_Z^-(U,V)=\operatorname{tr}\bigl(P_-(Z)^{-1}U^*Q_-(Z)^{-1}V\bigr),
\qquad g^-=\operatorname{Re}h^-,
\tag{L.12}
\]
is invariant, Kähler, complete and Hermitian symmetric. The domain is simply connected, with symmetry \(Z\mapsto-Z\) at zero.

**Proof.** The condition in (L.10) is equivalent to \(\|Z\|_{\mathrm{op}}<1\), where
\(\|Z\|_{\mathrm{op}}=\max_{|v|=1}|Zv|\); the maximum exists by compactness of the unit sphere and K.1 identifies its square with the largest eigenvalue of \(Z^*Z\). The triangle inequality follows by applying the vector triangle inequality to each unit vector. Consequently the unit ball is convex, bounded and open. Its straight-line contraction to zero proves simple connectivity. K.1 proves \(Q_->0\).

The graph \(E_Z\) is positive definite for the Hermitian form of signature \((p,q)\) exactly when \(P_->0\). If \(L=a+bZ\) had a nonzero kernel vector \(v\), the image under \(g\) of the positive-norm vector \((v,Zv)\) would have first component zero and hence nonpositive norm, a contradiction. Thus the denominator in (L.5) is invertible. Preservation of graph inner products gives
\[
P_-(W)=L^{-*}P_-(Z)L^{-1}>0.
\tag{L.13}
\]
The graph identity \(g\binom{I}{Z}=\binom{I}{W}L\) proves the action law by multiplication; the identity acts trivially, and inverses give inverse holomorphic transformations.

Here is an explicit element sending zero to \(Z\):
\[
g_Z=
\begin{pmatrix}
P_-^{-1/2}&Z^*Q_-^{-1/2}\\
ZP_-^{-1/2}&Q_-^{-1/2}
\end{pmatrix}.
\tag{L.14}
\]
Its diagonal block inner products for the form (L.11) are \(I_p,-I_q\), and the off-diagonal block is zero, so it belongs to \(U(p,q)\). This proves transitivity. If \(g\) fixes zero, then \(c=0\). The identities \(g^*Jg=J\) give \(a^*a=I_p\), \(a^*b=0\), hence \(b=0\) and \(d^*d=I_q\). This proves the stabilizer assertion.

The group is a closed Lie subgroup by Invariant connections A.1. It is connected as well: \(s\mapsto g_{sZ}\) is a path from \(I\) to \(g_Z\), and every \(g\) is \(g_{g\cdot0}k\) for some \(k\in U(p)\times U(q)\), which is connected by K.1. The quotient identification is a diffeomorphism, since (L.14) is a smooth global section for its orbit map and [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) gives its quotient charts.

Equation (L.6) still holds. Its second metric identity is now
\[
Q_-(W)=TQ_-(Z)T^*.
\tag{L.15}
\]
Indeed \(TZ=Wa-c\), so expanding gives
\[
\begin{aligned}
T(I-ZZ^*)T^*
&=dd^*-cc^*+W(bb^*-aa^*)W^*\\
&\quad+(-db^*+ca^*)W^*+W(-bd^*+ac^*)\\
&=I-WW^*.
\end{aligned}
\]
The final step is \(gJg^*=J\), which follows from \(g^*Jg=J\) by inversion. Positivity of \(Q_-(W)\) implies invertibility of \(T\). Equations (L.6), (L.13) and (L.15) give invariance of (L.12) by the same explicit cancellation as before. Positivity is the squared norm in (L.9) with minus subscripts. Conjugating the isometry \(Z\mapsto-Z\) by the transitive holomorphic action gives a holomorphic point symmetry everywhere. Theorem A.2 proves the Kähler condition and [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves completeness. □

**Proposition L.3 (Opposite brackets and exact curvature scale).** Under graph tangent coordinates at zero, the compact and noncompact metrics both have
\(g_0(U,V)=\operatorname{Re}\operatorname{tr}(U^*V)\). Their curvature operators are
\[
R_+(Z,W)V=ZW^*V-WZ^*V+VW^*Z-VZ^*W,\qquad
R_-=-R_+.
\tag{L.16}
\]
For a nonzero complex tangent vector \(Z\), their holomorphic sectional curvatures are
\[
H_\pm(Z)=
\frac{g_\pm(R_\pm(Z,iZ)iZ,Z)}{g_\pm(Z,Z)^2}
=\pm4\,
\frac{\operatorname{tr}((Z^*Z)^2)}
     {\operatorname{tr}(Z^*Z)^2}.
\tag{L.17}
\]
In particular they lie respectively in \([4/m,4]\) and \([-4,-4/m]\), where \(m=\min(p,q)\).

**Proof.** Conjugation by \(\operatorname{diag}(I_p,-I_q)\) is an involution of either acting group, with fixed subgroup the block-diagonal stabilizer. The odd tangent matrices are respectively
\[
X_+(Z)=\begin{pmatrix}0&-Z^*\\Z&0\end{pmatrix},
\qquad
X_-(Z)=\begin{pmatrix}0&Z^*\\Z&0\end{pmatrix}.
\tag{L.18}
\]
Differentiating the defining matrix-form equations gives these odd spaces and block-diagonal anti-Hermitian even parts. Conversely, any matrix satisfying the differentiated equation exponentiates into the group: the derivative of \((e^{tX})^*J e^{tX}\) is zero when \(X^*J+JX=0\), with \(J=I\) in the compact case and \(J=\operatorname{diag}(I_p,-I_q)\) in the other. The closed subgroup characterization in Invariant connections A.1 therefore proves equality of the stated Lie algebras. Differentiating the graph action at the identity sends either odd matrix to \(Z\). The odd brackets are opposite:
\[
\begin{aligned}
[X_+(Z),X_+(W)]
&=\operatorname{diag}(W^*Z-Z^*W,\;WZ^*-ZW^*),\\
[X_-(Z),X_-(W)]&=-[X_+(Z),X_+(W)].
\end{aligned}
\tag{L.19}
\]
The even action on the lower-left block is \(Z\mapsto dZ-Za\). The invariant metrics of L.1 and L.2 have the stated common value at zero, so [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) identifies their Levi-Civita connections with the canonical ones and gives \(R=-[[\,\cdot,\cdot\,],\cdot]\). Applying this to (L.19) proves (L.16), including its sign.

This also realizes the algebraic duality of D.3 explicitly. In the complexified compact algebra, replace its odd space by \(i\mathfrak m\); conjugation by \(\operatorname{diag}(I_p,-iI_q)\) fixes the even space and takes \(iX_+(Z)\) to \(X_-(Z)\). Restricting to total trace zero gives the corresponding pair of special-unitary algebras, with the same odd brackets. The scalar even centre acts trivially on the quotient and has no effect on these tangent computations.

Putting \(W=iZ\) and \(V=iZ\) in (L.16) gives \(R_+(Z,iZ)iZ=4ZZ^*Z\). Also \(Z\) and \(iZ\) are real-orthogonal and have equal norm, so the denominator defining sectional curvature is exactly the one in (L.17). This proves that formula. If the nonzero eigenvalues of \(Z^*Z\) are \(\lambda_1,\ldots,\lambda_r>0\), then \(r\leq m\) by K.1 and
\[
\frac1m\leq\frac1r
\leq\frac{\sum_j\lambda_j^2}{(\sum_j\lambda_j)^2}\leq1.
\]
The upper bound follows by expanding the squared sum. The lower bound follows from
\(r\sum_j\lambda_j^2-(\sum_j\lambda_j)^2=\sum_{j<k}(\lambda_j-\lambda_k)^2\geq0\).
Rank-one matrices attain the upper ratio, and rectangular diagonal matrices with \(m\) equal positive singular values attain the lower ratio. Thus the claimed bounds have their stated exact constants. □

**Example L.4 (The projective line and the disc).** For \(p=q=1\), the compact chart and the noncompact domain have metrics
\[
g_+=\frac{|dz|^2}{(1+|z|^2)^2},
\qquad
g_-=\frac{|dz|^2}{(1-|z|^2)^2},
\tag{L.20}
\]
of constant Gaussian curvature \(4\) and \(-4\), respectively. The Cayley map
\[
w=i\,\frac{1+z}{1-z}
\tag{L.21}
\]
maps the unit disc biholomorphically onto the upper half-plane and pulls its metric \(|dw|^2/(\operatorname{Im}w)^2\) back to \(4g_-\), of curvature \(-1\).

**Proof.** Formula (L.20) is (L.3) and (L.12) for scalar matrices. In complex dimension one the real tangent plane is the plane spanned by \(Z,iZ\), so its Gaussian curvature is the holomorphic sectional curvature in (L.17), namely \(4\) or \(-4\). Compactness is that of the full projective line, not of its single graph chart.

For (L.21), direct subtraction of its conjugate gives
\[
\operatorname{Im}w=\frac{1-|z|^2}{|1-z|^2},
\qquad
dw=\frac{2i\,dz}{(1-z)^2}.
\tag{L.22}
\]
Its inverse is \(z=(w-i)/(w+i)\); positivity of the imaginary part makes \(|w-i|<|w+i|\), proving that it lands in the disc. Both denominators are nonzero on their respective domains, so the two rational maps are inverse holomorphic maps. Substituting (L.22) proves the metric factor four. Multiplying a Riemannian metric by a positive constant leaves its Levi-Civita connection unchanged by the Koszul formula in [Symmetric spaces A.1](symmetric-spaces.md#theorem-a-1). Its sectional-curvature numerator scales by that constant and its denominator by its square; therefore curvature is divided by the constant. This gives curvature \(-1\) for \(4g_-\). □

## M. The Siegel upper half-space and its bounded coordinates

For an integer \(r\geq1\), let
\[
\mathcal H_r=\{W=X+iY: X,Y\in M_r(\mathbb R),\ X^T=X,\ Y^T=Y,\ Y>0\}.
\tag{M.1}
\]
It is an open convex subset of the complex vector space of symmetric matrices, of complex dimension \(r(r+1)/2\). Positivity is open by K.1; it is convex because each quadratic form remains positive under convex combinations. Its tangent vectors are complex symmetric matrices.

**Theorem M.1 (The symplectic action and the full stabilizer).** With
\[
\Omega=\begin{pmatrix}0&I_r\\-I_r&0\end{pmatrix},
\qquad
\mathrm{Sp}(2r,\mathbb R)=\{g\in GL(2r,\mathbb R):g^T\Omega g=\Omega\},
\tag{M.2}
\]
the formula
\[
g\cdot W=(AW+B)(CW+D)^{-1},
\qquad
g=\begin{pmatrix}A&B\\C&D\end{pmatrix},
\tag{M.3}
\]
is a transitive holomorphic action on \(\mathcal H_r\). The group is connected, and the stabilizer of \(iI_r\) is
\[
K=\left\{\begin{pmatrix}A&B\\-B&A\end{pmatrix}: A-iB\in U(r)\right\}
\cong U(r).
\tag{M.4}
\]
The resulting identification \(\mathrm{Sp}(2r,\mathbb R)/K\cong\mathcal H_r\) is a diffeomorphism.

**Proof.** Write \(F_W=\binom{W}{I_r}\). Since \(W^T=W\) and \(Y>0\),
\[
F_W^T\Omega F_W=0,\qquad F_W^*\Omega F_W=-2iY.
\tag{M.5}
\]
If \(N=CW+D\) had a nonzero kernel vector \(v\), then \(gF_Wv\) would have lower block zero and hence have zero pairing with itself under \(\Omega\). But the second identity in (M.5), and reality of \(g\), make that pairing \(-2i\,v^*Yv\ne0\). Thus \(N\) is invertible.

Set \(W'=(AW+B)N^{-1}\). The graph equation
\[
gF_W=F_{W'}N
\tag{M.6}
\]
and the first identity in (M.5) imply \(W'^T=W'\). The second identity then gives
\[
Y'=\operatorname{Im}W'=N^{-*}YN^{-1}>0.
\tag{M.7}
\]
So (M.3) maps the domain into itself. Equation (M.6), applied twice, proves the action law because matrix multiplication is associative and the normalized graph has unique upper block. The identity acts trivially; the inverse matrix gives the inverse action. Its entries are holomorphic rational functions with nonvanishing denominator, proving holomorphicity.

For any \(W=X+iY\), let \(S=Y^{1/2}\), the real positive square root supplied by K.1, and put
\[
g_W=\begin{pmatrix}S&XS^{-1}\\0&S^{-1}\end{pmatrix}.
\tag{M.8}
\]
Multiplying the blocks shows \(g_W^T\Omega g_W=\Omega\): the upper-right relation is \(S^TS^{-1}=I\), the lower-right skew block vanishes because \(X=X^T\), and the other relations are their transposes. Its action sends \(iI_r\) to \(X+iY\). This proves transitivity and gives a smooth global section.

Fixing \(iI_r\) in (M.3) means \(iA+B=-C+iD\), hence \(D=A\), \(C=-B\). Substitution in the symplectic equations gives
\[
A^TA+B^TB=I,\qquad A^TB=B^TA.
\tag{M.9}
\]
These are exactly \((A-iB)^*(A-iB)=I\). The real block matrix in (M.4) is the real representation of the complex matrix \(A-iB\), so this identification preserves multiplication and gives the full stabilizer, not only its identity component.

The symplectic group is a closed matrix Lie subgroup by Invariant connections A.1. It is connected: the segment
\(W_t=tX+i((1-t)I+tY)\) stays in \(\mathcal H_r\), and (M.8) supplies a path from the identity to \(g_W\). Every \(g\) equals \(g_{g\cdot iI}k\) with \(k\in K\), and K.1 makes \(K\) connected. Finally [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) gives the quotient manifold. The orbit map descends smoothly to the quotient; its inverse is \(W\mapsto g_WK\), smooth by (M.8). They are inverse by transitivity and the stabilizer calculation. □

**Theorem M.2 (The metric and global point symmetries).** The Hermitian form
\[
\begin{aligned}
h_W(U,V)&=\operatorname{tr}(Y^{-1}\bar UY^{-1}V),\\
g_W(U,V)&=\operatorname{Re}h_W(U,V),
\qquad U^T=U,\ V^T=V,
\end{aligned}
\tag{M.10}
\]
is positive and invariant under (M.3). It is Kähler and makes \(\mathcal H_r\) a complete simply connected Hermitian symmetric space. Its point symmetry at \(iI_r\) is \(W\mapsto-W^{-1}\). At \(W_0=X_0+iY_0\) its point symmetry is
\[
s_{W_0}(W)=X_0-Y_0(W-X_0)^{-1}Y_0.
\tag{M.11}
\]

**Proof.** Because \(U\) is symmetric, \(U^*=\bar U\). Hence
\[
h_W(U,U)=\|Y^{-1/2}UY^{-1/2}\|_{\mathrm{HS}}^2>0
\quad(U\ne0).
\tag{M.12}
\]
Conjugate symmetry follows from taking the conjugate transpose inside the trace, and complex linearity in the second variable and conjugate linearity in the first are immediate. Thus \(g\) is a smooth positive Hermitian metric with exactly the scale in (M.10).

Let \(N=CW+D\) as in M.1. Differentiation of (M.3) gives
\[
d(g\cdot)_W(U)=(A-W'C)UN^{-1}=N^{-T}UN^{-1}.
\tag{M.13}
\]
For the last equality, multiply \(A-W'C\) on the left by \(N^T\). Symmetry of \(W'\) and \(W'N=AW+B\) give
\[
N^T(A-W'C)
=(WC^T+D^T)A-(WA^T+B^T)C=I.
\]
Here \(C^TA=A^TC\) and \(D^TA-B^TC=I\) are the corresponding blocks of \(g^T\Omega g=\Omega\).

The real symmetric matrix \(Y'\) in (M.7) gives two equal expressions for its inverse:
\[
Y'^{-1}=NY^{-1}\bar N^T
       =\bar NY^{-1}N^T.
\tag{M.14}
\]
Use the first expression before \(\overline{d(g\cdot)U}\), and the second before \(d(g\cdot)V\). Equations (M.13)–(M.14) yield
\[
\begin{aligned}
&Y'^{-1}\overline{d(g\cdot)U}\,Y'^{-1}d(g\cdot)V\\
&\quad=
(NY^{-1}\bar N^T)(\bar N^{-T}\bar U\bar N^{-1})
(\bar NY^{-1}N^T)(N^{-T}VN^{-1})\\
&\quad=NY^{-1}\bar UY^{-1}VN^{-1}.
\end{aligned}
\tag{M.15}
\]
Taking traces proves invariance, without interchanging noncommuting matrix factors.

The symplectic matrix \(\begin{pmatrix}0&-I\\I&0\end{pmatrix}\) acts by \(s(W)=-W^{-1}\). The inverse exists: if \(Wv=0\), then \(\operatorname{Im}(v^*Wv)=v^*Yv>0\) for \(v\ne0\), a contradiction. This holomorphic isometry fixes \(iI\); its differential is \(U\mapsto W^{-1}UW^{-1}\), equal to \(-U\) at \(iI\). Conjugate it by the transitive action in M.1 to obtain point symmetries everywhere.

More explicitly the map associated to (M.8) for \(W_0\) is
\(F(Q)=X_0+S_0QS_0\), with \(S_0=Y_0^{1/2}\). Its inverse is
\(F^{-1}(W)=S_0^{-1}(W-X_0)S_0^{-1}\). Substituting into \(F\circ s\circ F^{-1}\) proves (M.11). The inverse of \(W-X_0\) exists by the same imaginary-part argument, so this formula is global.

Theorem A.2 makes a Hermitian metric with these holomorphic point symmetries Kähler. [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves its geodesic completeness. Convexity from (M.1) contracts every loop by the straight-line homotopy to \(iI\); thus the space is simply connected. None of these conclusions requires an external integrability or completeness theorem. □

**Proposition M.3 (The symmetric matrix ball and its factor four).** The transformations
\[
Z=(W-iI)(W+iI)^{-1},\qquad
W=i(I+Z)(I-Z)^{-1}
\tag{M.16}
\]
are inverse biholomorphisms between \(\mathcal H_r\) and
\[
\mathcal B_r=\{Z\in M_r(\mathbb C):Z^T=Z,\ I-Z^*Z>0\}.
\tag{M.17}
\]
Under these coordinates the metric (M.10) is
\[
h^{\mathcal B}_Z(U,V)
=4\operatorname{tr}\bigl((I-Z^*Z)^{-1}U^*(I-ZZ^*)^{-1}V\bigr),
\quad U^T=U,\quad V^T=V.
\tag{M.18}
\]
The symmetry at the origin is \(Z\mapsto-Z\). For \(r=1\) this is the curvature-\(-1\) metric of L.4.

**Proof.** For \(W\in\mathcal H_r\), \(W+iI\) is invertible by positivity of its imaginary part, as in M.2. The resulting \(Z\) is symmetric: the two polynomials in \(W\) commute with each other and with their inverses, so transposition leaves their product unchanged. Moreover
\[
\begin{aligned}
&(W+iI)^*(I-Z^*Z)(W+iI)\\
&\qquad=(W+iI)^*(W+iI)-(W-iI)^*(W-iI)=4Y>0.
\end{aligned}
\tag{M.19}
\]
This proves membership in (M.17).

Conversely, if \(Z\in\mathcal B_r\), then \(I-Z\) is invertible: \(Zv=v\ne0\) would contradict \(v^*(I-Z^*Z)v>0\). Set \(A=I-Z\), \(P=I-Z^*Z\), \(Q=I-ZZ^*\). Transposition again proves that the inverse formula in (M.16) is symmetric. Direct subtraction of its conjugate transpose gives
\[
Y=A^{-*}PA^{-1}>0.
\tag{M.20}
\]
Indeed, after multiplication by \(A^*\) and \(A\), the sum
\(A^*(I+Z)+(I+Z^*)A\) is \(2P\). Since \(W\) is symmetric, its imaginary part is real symmetric, so this is a point of \(\mathcal H_r\). Solving the two elementary linear-fractional equations proves that the maps in (M.16) are inverse. Their rational entries are holomorphic on the domains just verified.

For completeness compute the metric, not only the complex structure. Differentiating the inverse map gives
\[
dW(U)=2iA^{-1}UA^{-1}.
\tag{M.21}
\]
To obtain this, differentiate \(i(I+Z)A^{-1}\), then use
\(I+(I+Z)A^{-1}=2A^{-1}\). Since \(Z\) is symmetric, \(A^*=\bar A\), \(P^T=Q\); (M.20) and reality of \(Y\) imply
\[
Y^{-1}=AP^{-1}\bar A=\bar A Q^{-1}A.
\tag{M.22}
\]
Insert (M.21) and its conjugate in (M.10), using the first expression in (M.22) before the conjugate tangent and the second before the other tangent. The product inside the trace reduces to
\[
4AP^{-1}\bar UQ^{-1}VA^{-1}.
\]
Since \(U^*=\bar U\), cyclicity gives (M.18). Substitution of \(-W^{-1}\) into the first map of (M.16) gives \(-Z\), or equivalently substitution of \(-Z\) into its inverse gives \(-W^{-1}\). At zero (M.18) is \(4\operatorname{tr}(U^*V)\), explaining its factor four relative to the matrix-ball metric (L.12) restricted to symmetric tangents. Its scalar case and curvature are exactly those of L.4, with the map and metric calculation (L.21)–(L.22). □

## N. A fully normalized Siegel exercise

**Exercise N.1 (Action, metric and reflection).** Verify the Hermitian symmetric structure on \(\mathcal H_r\), including the domain of the action, its stabilizer, the scale of the metric, and a global point symmetry. State the resulting disc metric when \(r=1\).

**Solution.** Use the alternating form \(\Omega\) and graph column \(F_W\) from (M.2) and (M.5). Their identities prove both invertibility of \(CW+D\) and positivity of
\[
\operatorname{Im}(g\cdot W)=(CW+D)^{-*}Y(CW+D)^{-1}.
\]
The graph multiplication (M.6) proves the action law, and the section (M.8) sends \(iI\) to every \(X+iY\). Solving the stabilizer equations gives \(D=A,\ C=-B\) and \(A-iB\in U(r)\), so it is the full group in (M.4).

At \(iI\) choose exactly
\[
h_{iI}(U,V)=\operatorname{tr}(\bar U V).
\]
The derivative and the two inverse-imaginary-part formulas (M.13)–(M.15) prove that its invariant extension is (M.10), with no additional scalar. In particular, for symmetric \(U\),
\[
h_{iI}(U,U)=\sum_a|U_{aa}|^2+2\sum_{a<b}|U_{ab}|^2.
\tag{N.1}
\]
The factor two on off-diagonal symmetric coordinates comes from counting both entries in the trace; replacing this by the unweighted norm of the independent upper-triangular entries would give a different form.

The matrix \(\begin{pmatrix}0&-I\\I&0\end{pmatrix}\) gives the symmetry \(W\mapsto-W^{-1}\) at \(iI\), with derivative \(-I\); (M.11) gives the global symmetry at \(X_0+iY_0\). Invariance makes these maps isometries, and rationality makes them holomorphic. A.2 and [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) supply the Kähler condition and completeness. Convexity contracts loops, proving simple connectivity.

Finally the Cayley transform has differential \(dW=2i\,dZ\) at zero. Thus the upper-half-space metric corresponds at zero to \(4\operatorname{tr}(U^*V)\) on the bounded model; M.3 proves the global formula (M.18). For \(r=1\) this is
\[
\frac{4|dz|^2}{(1-|z|^2)^2}
\quad\longleftrightarrow\quad
\frac{|dw|^2}{(\operatorname{Im}w)^2},
\]
both of curvature \(-1\) by L.4. This verifies the action, geometry and every stated normalization. □

## O. Positive Cartan matrices

We next classify a finite matrix object before relating it to roots. A **positive symmetrizable Cartan matrix** is an integral \(n\times n\) matrix \(A\), \(n\geq1\), such that
\[
a_{ii}=2,\qquad a_{ij}\leq0\ (i\ne j),\qquad
a_{ij}=0\Longleftrightarrow a_{ji}=0,
\tag{O.1}
\]
and a positive diagonal matrix \(D=\operatorname{diag}(d_i)\) makes \(DA\) symmetric positive definite. Its graph joins \(i,j\) when \(a_{ij}\ne0\). Connectedness is equivalent to indecomposability under simultaneous row and column permutation: a partition with no crossing edges is exactly a diagonal block decomposition.

Our convention in this part and the next is
\[
a_{ij}=\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.
\tag{O.2}
\]
The denominator belongs to the row. This transposes the column convention in [De Rham decomposition AE.1](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-1). We retain the geometric arrow convention: a multiple edge points from a longer vector to a shorter one.

**Lemma O.1 (normalization and length data).** Put \(k_{ij}=a_{ij}a_{ji}\) for distinct vertices. Then \(k_{ij}\in\{0,1,2,3\}\), and
\[
Q=D^{-1/2}(DA)D^{-1/2},\qquad
Q_{ii}=2,\quad Q_{ij}=-\sqrt{k_{ij}}
\tag{O.3}
\]
is positive definite. There are independent vectors \(\alpha_i\) with Gram matrix \(DA\); they satisfy (O.2) and \(\|\alpha_i\|^2=2d_i\). Along an edge,
\[
\frac{d_j}{d_i}=\frac{a_{ij}}{a_{ji}}.
\tag{O.4}
\]
Consequently the symmetrizer is unique up to multiplication by a positive constant on each connected component. An edge has multiplicity \(k_{ij}\); together with the arrow toward smaller \(d_i\), this determines its two Cartan entries.

**Proof.** Symmetry gives \(d_i a_{ij}=d_j a_{ji}\). Congruence by the invertible diagonal matrix \(D^{-1/2}\) preserves positivity, since its quadratic form at \(x\ne0\) is that of \(DA\) at \(D^{-1/2}x\ne0\). For an edge both entries are negative, so its off-diagonal entry is
\[
\sqrt{d_i/d_j}\,a_{ij}=-\sqrt{a_{ij}a_{ji}}.
\]
Restriction to the two coordinate directions has quadratic form
\[
2x^2-2\sqrt{k_{ij}}xy+2y^2
=2\bigl(x-\tfrac12\sqrt{k_{ij}}y\bigr)^2
+\bigl(2-\tfrac12k_{ij}\bigr)y^2.
\]
Thus \(k_{ij}<4\). It is a nonnegative integer, proving the list.

Give \(\mathbb R^n\) the inner product \(x^tDAy\), and take its coordinate basis for \(\alpha_i\). This constructs the vectors directly and verifies (O.2). Equation (O.4) propagates the ratios of all \(d_i\) along paths. It proves the uniqueness assertion, including singleton components.

On a single edge the entries are \((-1,-1)\). On a double or triple edge their absolute values have product two or three, so one is one and the other is respectively two or three. Equation (O.4) puts the larger absolute entry in the row of the shorter vector. Thus for vertices ordered long then short the entries are \((-1,-2)\) or \((-1,-3)\). This also proves that neither the multiplicities nor the arrows depend on the chosen symmetrizer. □

**Theorem O.2 (the complete positive Cartan list).** Up to simultaneous permutation of rows and columns, the connected matrices of (O.1) are exactly the following:

| Type | Graph and row convention |
|---|---|
| \(A_n,\ n\geq1\) | A chain of \(n\) vertices, all edges single |
| \(B_n,\ n\geq2\) | The chain \(1,\ldots,n\), last edge double, with \((a_{n-1,n},a_{n,n-1})=(-1,-2)\) |
| \(C_n,\ n\geq2\) | The same chain, with last pair \((-2,-1)\) |
| \(D_n,\ n\geq4\) | One branch, with the single-edge arm triple \((1,1,n-3)\) |
| \(E_6,E_7,E_8\) | One branch, with single-edge arm triples \((1,2,2)\), \((1,2,3)\), \((1,2,4)\), respectively |
| \(F_4\) | The chain \(1,2,3,4\), middle edge double, with \((a_{23},a_{32})=(-1,-2)\) |
| \(G_2\) | Two vertices, with \((a_{12},a_{21})=(-1,-3)\) |

All diagonal entries are two, both entries on a single edge are minus one, and entries outside the edges are zero. Arm lengths count edges from the branch. The two names \(B_2,C_2\) describe the same matrix up to exchanging vertices; for \(n\geq3\) the \(B_n,C_n\) types are distinct. Reversing a path identifies the two arrow orientations in each of \(F_4,G_2\). Apart from the stated rank-two duplication, the table has no repetitions.

**Proof.** By O.1, \(Q\) is twice the Gram matrix of independent unit vectors, with off-diagonal entries \(-\sqrt{k}/2\) in that Gram matrix. The restrictions proved in [De Rham decomposition AE.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-ae-2) use exactly these facts about the Gram matrix; they do not use reflection closure or any other root axiom. They therefore apply to the vectors just constructed, even though no root system has yet been specified. Here are the complete restrictions and their positivity tests.

The graph is a tree: coefficients one on a cycle with \(m\) vertices give quadratic value at most \(2m-2m=0\), with any extra edges decreasing it. On the star of a vertex, whose neighbours are pairwise unjoined, completing the neighbour squares gives central coefficient
\[
2-\frac12\sum_{j\sim i}k_{ij}>0.
\tag{O.5}
\]
Thus the sum is at most three. The degree is at most three; three incident edges must all be single; a triple edge fills the bound at both ends and is an isolated \(G_2\) component.

The three remaining forbidden configurations have the explicit nonpositive test vectors of [De Rham decomposition AE.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-ae-2), equations (AE.8)–(AE.10). We spell them out to make the matrix scope clear. Between two branch vertices choose a shortest path with \(m\) vertices, assign two to each path vertex and one to two further neighbours of each endpoint. Its quadratic value is at most
\[
(8m+8)-8(m-1)-16=0.
\]
The extra neighbours are distinct because the graph is a tree. Two double edges would give a path with double end edges and single intervening edges; adjacent double edges are already excluded by (O.5). Give its outer endpoints coefficient one and its interior vertices coefficient \(\sqrt2\). For \(m\) vertices the value is
\[
(4m-4)-8-4(m-3)=0.
\]
If a branch and a double edge coexist, follow a path from the branch through the first double edge to its far endpoint. Give the \(m\) vertices before that endpoint coefficient two, the far endpoint \(\sqrt2\), and two other neighbours of the branch coefficient one. The value is
\[
(8m+8)-8(m-1)-8-8=0.
\]
All contradict positivity. Hence there is at most one branch or one double edge, and they cannot coexist.

To bound the possible lengths, let \(T_l\) be the single-edge chain matrix with diagonal two. The calculations in the same earlier proof are
\[
\begin{aligned}
x^tT_lx&=x_1^2+x_l^2+\sum_{i=1}^{l-1}(x_i-x_{i+1})^2>0
       &&(x\ne0),\\
(T_l^{-1})_{11}&=\frac{l}{l+1}.
\end{aligned}
\tag{O.6}
\]
For \(l=1\), the first expression means \(2x_1^2\). To check the inverse entry, substitute \(z_i=(l+1-i)/(l+1)\) in \(T_lz=e_1\). Completing squares in a block matrix gives
\[
\begin{aligned}
d t^2+2t b^ty+y^tBy
={}&(y+B^{-1}bt)^tB(y+B^{-1}bt)\\
&+(d-b^tB^{-1}b)t^2.
\end{aligned}
\tag{O.7}
\]
For three arms \(1\leq p\leq q\leq r\), eliminate their positive chain blocks with (O.6)–(O.7). The central coefficient is positive exactly when
\[
\frac1{p+1}+\frac1{q+1}+\frac1{r+1}>1.
\tag{O.8}
\]
If \(p\geq2\), the left side is at most one. Thus \(p=1\). When \(q=1\), every \(r\geq1\) works, giving \(D_{r+3}\). When \(q\geq3\), the sum is at most \(1/2+1/4+1/4\). The remaining case \(q=2\) requires \(r<5\), so \(r=2,3,4\), giving the three \(E\) types.

For a chain with one double edge, removing it gives blocks with \(p,q\geq1\) vertices. Eliminate the first block. The second form is
\[
y^tT_qy-\frac{2p}{p+1}y_1^2.
\]
For completeness put \(u=T_q^{-1}e_1\), \(a=u_1=q/(q+1)\), and \(y=tu+z\) with \(z_1=0\), \(t=y_1/a\). Since \(u^tT_qz=0\), this form becomes
\[
z^tT_qz+t^2a\left(1-\frac{2p}{p+1}a\right).
\]
It is positive definite exactly when
\[
1-\frac{2pq}{(p+1)(q+1)}>0
\quad\Longleftrightarrow\quad(p-1)(q-1)<2.
\tag{O.9}
\]
The possibilities are \(p=1\), \(q=1\), or \(p=q=2\). They give the end-double chains \(B,C\) or \(F_4\). A tree with no branch and no multiple edge is \(A_n\).

These tests also prove existence for every matrix in the table, without assuming a root realization. For a simply laced graph take all \(d_i=1\). For a multiple edge assign \(d_i=2\) on its long side and one on its short side, or three and one for \(G_2\); every other edge is single and propagates its endpoint value unchanged. Then \(DA\) is symmetric. Its normalized matrix is positive by (O.6), (O.8) or (O.9); for \(G_2\) the two-coordinate square completion in O.1 is positive with \(k=3\). Congruence gives positivity of \(DA\).

Finally the tree shape, arm lengths, location of the multiple edge and its orientation recover the table entries uniquely. The two ends of the rank-two chain can be exchanged. For longer end-double chains, the vertex at the double edge which is an endpoint is intrinsic: its being short or long distinguishes \(B_n\) from \(C_n\). In \(F_4\) and \(G_2\), path reversal exchanges the drawings. For branch types, the unordered arm triple gives exactly the ranks and types stated. This proves both exhaustion and the claimed identifications. □

## P. Roots, changes of simple basis and classification

A **reduced crystallographic root system** in a Euclidean space \(V\) is a finite spanning set \(R\) of nonzero vectors such that
\[
s_\alpha(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha
\tag{P.1}
\]
permutes \(R\) for every \(\alpha\in R\), every number \(2(\beta,\alpha)/(\alpha,\alpha)\) is integral, and the only roots on the line through \(\alpha\) are \(\alpha,-\alpha\). A **simple basis** is a basis \(\Delta=(\alpha_1,\ldots,\alpha_n)\) in \(R\) such that every root has integer coefficients all nonnegative or all nonpositive in this basis. A vector is **regular** if its inner product with every root is nonzero. We use the row Cartan convention (O.2).

**Theorem P.1 (realization and uniqueness with a specified basis).** Every positive symmetrizable Cartan matrix, connected or not, is the row Cartan matrix of a reduced crystallographic root system with an ordered simple basis. For a specified symmetrizer \(D\), the realization can be chosen with Gram matrix \(DA\). Any two realizations with the same ordered Cartan matrix are related by the unique linear map taking one simple basis to the other; this map is a similarity on each connected component.

For every root system a simple basis exists. The sign-coefficient definition above is equivalent to being the indecomposable positive roots for some regular vector. If \(W\) is generated by the reflections in a simple basis, then
\[
R=\bigcup_{i=1}^n W\alpha_i.
\tag{P.2}
\]
The graph components give precisely the orthogonal irreducible components of \(R\).

**Proof.** [De Rham decomposition AE.1](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-1) proves from the abstract root axioms that, for each regular vector, the positive roots which cannot be split into two positive roots form a basis, and that every positive root has a nonnegative integral expansion in it. Its proof explicitly supplies the root-difference argument for abstract systems, so this result needs no Lie-algebra origin. The existence of a regular vector is also proved there using the finite-hyperplane argument in [De Rham decomposition AD.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ad-5).

Conversely, let \(\Delta\) be a basis with the sign-coefficient property. Solve \((v,\alpha_i)=1\) for \(v\): the Gram matrix is invertible because its quadratic form is the squared norm of a nonzero linear combination of a basis. Every root then pairs with \(v\) as a nonzero integer of its coefficient sign, so \(v\) is regular. A basis root cannot split into two positive roots, since its unit coefficient vector cannot be the sum of two nonzero nonnegative integer vectors. The indecomposable positive roots form a basis by the preceding programme result and already contain the \(n\) members of \(\Delta\). They therefore equal \(\Delta\). This proves the equivalence and makes the orbit and component conclusions of [De Rham decomposition AE.1](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-1) applicable to every simple basis as defined here. In particular that proof's height descent gives (P.2).

Now take a matrix \(A\). Its connected blocks are on the list of O.2. Complete coordinate realizations with all root axioms and all simple-basis sign checks are proved in [De Rham decomposition AE.3](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-3) for \(A,B,C,D\), AE.4 for \(E_6,E_7,E_8\), and AE.5 for \(F_4,G_2\). Transpose their recorded column Cartan matrices to obtain our row convention. The \(B,C\) coordinates then have exactly the end entries of O.2. The \(F_4\) basis has its first two roots long and its last two short, giving \(a_{23}=-1,a_{32}=-2\). For \(G_2\), exchange the earlier basis order, which was short then long, to obtain our long-first order and entries \(-1,-3\). Use the \(B_2\) realization with exchanged labels for \(C_2\). All further vertex permutations in O.2 are implemented by reordering the same basis.

Take the orthogonal direct sum of these Euclidean spaces, with root set the union of their root sets in their respective summands. It is finite, spans, and is reduced. A reflection in one summand acts identically on every other summand and permutes the roots in its own; cross-summand Cartan integers are zero. The union of the simple bases is a basis, and each root retains its one-sign expansion. Thus this is the required root system for every block matrix.

If \(d_i^0=\|\alpha_i\|^2/2\) in this realization, then \(\operatorname{diag}(d_i^0)A\) is its Gram matrix. O.1 implies that \(d_i/d_i^0=c>0\) is constant on each connected component. Multiply every vector in that component by \(\sqrt c\). Reflections and Cartan integers are unchanged, and the new Gram matrix is precisely \(DA\).

For uniqueness let \(T\alpha_i=\alpha_i'\) between two realizations with the same ordered matrix. The reflection formula is
\[
s_i\alpha_j=\alpha_j-a_{ij}\alpha_i.
\tag{P.3}
\]
It shows that \(T\) intertwines the corresponding simple reflections. Applying (P.2) gives \(T(R)=R'\). Along each graph edge the squared-length ratios are fixed by (O.4), so the two Gram matrices differ by a positive scalar on each connected component. Entries between components are zero. Thus \(T\) is a componentwise similarity.

For clarity the component assertion can also be read directly from (P.2): distinct component spans are orthogonal, each generating reflection fixes all other spans, and each root is in one component span. Those intersections are root systems. An orthogonal splitting of one of them would partition its simple roots into unjoined sets, contrary to connectedness. Conversely every orthogonal decomposition partitions the simple roots in just this way. This proves the asserted uniqueness of the irreducible components. □

**Theorem P.2 (independence of the simple basis).** Let \(\Delta,\Delta'\) be two simple bases of \(R\). The group \(W\) generated by the reflections in \(\Delta\) contains an element sending \(\Delta\) to \(\Delta'\) as unordered sets. Hence the two ordered Cartan matrices differ only by simultaneous row and column permutation, and their directed diagrams agree.

**Proof.** Every generator is an isometry permuting the finite spanning set \(R\). The map from \(W\) to its permutations is injective: a linear transformation fixing every root fixes their span. Thus \(W\) is finite.

Choose \(u,v\) by \((u,\alpha_i)=1\) for \(\Delta\) and \((v,\alpha_i')=1\) for \(\Delta'\). P.1 proves that these equations are solvable and that both vectors are regular, with positive systems determined by the two bases. Maximize \((wv,u)\) over \(w\in W\), possible by finiteness. If \((wv,\alpha_i)<0\), then
\[
(s_iwv,u)-(wv,u)
=-\frac{2(wv,\alpha_i)(u,\alpha_i)}{(\alpha_i,\alpha_i)}>0,
\]
contradicting the maximum. Equality \((wv,\alpha_i)=0\) is impossible: \(w^{-1}\alpha_i\) is a root and \(v\) is regular. Consequently \(wv\) pairs positively with every member of \(\Delta\).

The one-sign expansions imply that \(wv\) and \(u\) define exactly the same positive roots. Since \(w\) is an isometry,
\[
\{\beta\in R:(wv,\beta)>0\}
=w\{\gamma\in R:(v,\gamma)>0\}.
\]
A linear bijection preserving these positive sets preserves whether a root is a sum of two positive roots. By the equivalence in P.1, their indecomposable roots are respectively \(\Delta\) and \(w\Delta'\). Thus \(w\Delta'=\Delta\); its inverse is the required element. This finite-orbit maximization is the same elementary mechanism used for the reversing element in [De Rham decomposition AD.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ad-5).

An isometry preserves every ratio in (O.2). Reordering the basis only permutes its indices simultaneously. The matrix and diagram conclusions follow. □

**Corollary P.3 (classification without a chosen basis).** Up to permutation of components and a positive scale on each component, reduced crystallographic root systems are classified by finite multisets of connected diagrams
\[
A_n\ (n\geq1),\quad B_n\ (n\geq2),\quad C_n\ (n\geq3),
\quad D_n\ (n\geq4),\quad E_6,E_7,E_8,F_4,G_2.
\tag{P.4}
\]
The omitted name \(C_2\) is \(B_2\). If rank zero is allowed, its unique root system is \(R=\varnothing\) in \(V=0\), corresponding to the empty multiset.

**Proof.** Choose a simple basis using P.1. Its row Cartan matrix satisfies (O.1): the nonpositive off-diagonal entries and integrality are proved in [De Rham decomposition AE.1](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-1), with the denominator convention transposed. For \(d_i=\|\alpha_i\|^2/2\), its symmetrization is the positive Gram matrix. Apply O.2 to its connected blocks. P.1 identifies those blocks with the intrinsic orthogonal irreducible components, and P.2 shows that the diagram multiset is independent of the basis.

Every such multiset is realized by the direct-sum construction in P.1. If two multisets agree, choose and order bases with the same block Cartan matrix. P.1 gives a similarity on each block taking every root to a root, so the systems are equivalent.

Conversely an equivalence sends a simple basis to a simple basis: root coefficient expansions are preserved by the linear map, and positive component scales preserve all the Cartan ratios. The resulting matrices agree in the transported order. Applying P.2 to any other choice of basis proves equality of the diagram multisets. O.2 proves that the only duplicated name in its original table was \(B_2=C_2\), so (P.4) has no further identifications. In dimension zero, the absence of nonzero vectors forces the stated empty system. This completes the classification of these root systems. □

This classification concerns Euclidean root systems. A classification of Lie algebras or of Hermitian symmetric pairs requires the additional constructions connecting those objects with roots.

## Q. Roots of an arbitrary complex semisimple algebra

In this part \(\mathfrak g\) is a finite-dimensional **complex** semisimple Lie algebra. Semisimplicity means the absence of nonzero solvable ideals. Its Killing form is
\[
K(X,Y)=\operatorname{tr}_{\mathfrak g}(\operatorname{ad}X\,\operatorname{ad}Y).
\tag{Q.1}
\]
The complete algebraic foundations are in [Symmetric spaces H.1–H.6](symmetric-spaces.md#lemma-h-1): polynomial Jordan decomposition, Lie triangularization, Engel's theorem, the Killing criterion and inner derivations. Complete reducibility for all finite-dimensional semisimple modules is proved in [Symmetric spaces I.3](symmetric-spaces.md#corollary-i-3). No compact real form is assumed here.

**Lemma Q.1 (all finite-dimensional rank-one modules).** Suppose endomorphisms \(E,F,H\) of a finite-dimensional complex vector space satisfy
\[
[H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=H.
\tag{Q.2}
\]
Their representation is a direct sum of irreducible strings \(V_m\), \(m\in\mathbb Z_{\geq0}\). Each has basis \(v_0,\ldots,v_m\) with
\[
\begin{aligned}
Hv_j&=(m-2j)v_j,\\
Fv_j&=v_{j+1}\quad(j<m),& Fv_m&=0,\\
Ev_0&=0,& Ev_j&=j(m-j+1)v_{j-1}\quad(j\geq1).
\end{aligned}
\tag{Q.3}
\]
In particular \(H\) is diagonalizable with integer eigenvalues, and \(E,F\) are nilpotent. Every irreducible string has one-dimensional weight spaces and is uniquely determined up to isomorphism by \(m\).

**Proof.** The trace-zero two-by-two matrices have basis
\[
e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad
f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad
h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\]
Their matrix commutators are (Q.2). The adjoint matrices give
\[
K(h,h)=8,\qquad K(e,f)=K(f,e)=4,
\]
with all other basis pairings zero. Indeed \(\operatorname{ad}h\) has eigenvalues \(2,-2,0\); the product \(\operatorname{ad}e\,\operatorname{ad}f\) has diagonal coefficients \(2,0,2\) on \(e,f,h\), respectively. The other traces follow by the same three commutators. The resulting Gram determinant is \(-128\), so [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) proves that this algebra \(\mathfrak{sl}_2(\mathbb C)\) is semisimple.

Relations (Q.2) define a representation of this three-dimensional algebra. [Symmetric spaces I.3](symmetric-spaces.md#corollary-i-3) decomposes every such module into irreducible submodules. It remains to determine an irreducible nonzero one.

By [Symmetric spaces H.1](symmetric-spaces.md#lemma-h-1), \(H\) has a nonempty finite eigenvalue set. There is an eigenvalue \(\lambda\) for which \(\lambda+2\) is absent: starting from any eigenvalue and repeatedly adding two must eventually leave that finite set. Choose an eigenvector \(v_0\) at such a \(\lambda\). The first commutator makes \(Ev_0\) an eigenvector at \(\lambda+2\) if it is nonzero, so \(Ev_0=0\). Put \(v_j=F^jv_0\). Induction gives
\[
Hv_j=(\lambda-2j)v_j,\qquad
Ev_j=j(\lambda-j+1)v_{j-1}.
\tag{Q.4}
\]
For the second identity commute \(E\) past the first \(F\) using \(EF=FE+H\). The coefficient becomes
\((j-1)(\lambda-j+2)+\lambda-2j+2=j(\lambda-j+1)\).

The nonzero \(v_j\) have distinct eigenvalues and hence are independent, by the polynomial projections in [Symmetric spaces H.1](symmetric-spaces.md#lemma-h-1). There is a last nonzero \(v_m\). The next identity in (Q.4) says
\[
0=(m+1)(\lambda-m)v_m,
\]
so \(\lambda=m\). All earlier vectors are nonzero, since one zero would make every subsequent \(F\)-iterate zero. Their span is invariant under all three operators and is therefore the entire irreducible module. This gives (Q.3).

Conversely (Q.3) defines operators satisfying (Q.2), as substitution at each \(v_j\), including \(j=0,m\), verifies. The resulting module is irreducible: an invariant subspace is preserved by the polynomial projections onto the distinct \(H\)-eigenlines. A nonzero subspace thus contains some \(v_j\); repeated \(E\), whose coefficients before \(v_0\) are nonzero, reaches \(v_0\), and repeated \(F\) reaches every basis vector. The displayed matrices also give uniqueness. The direct-sum decomposition now proves all assertions for arbitrary modules, including the zero module. □

An element of \(\mathfrak g\) is called **semisimple** when its adjoint operator is diagonalizable, and **nilpotent** when that operator is nilpotent. A **toral subalgebra** is an abelian subalgebra of semisimple elements. Here a **Cartan subalgebra** means a toral subalgebra equal to its centralizer. A Cartan is automatically maximal toral, since any larger toral subalgebra would centralize it.

**Theorem Q.2 (Cartan existence and the root decomposition).** Every maximal toral subalgebra \(\mathfrak h\) is a Cartan subalgebra. Such subalgebras exist; if \(\mathfrak g\ne0\), they are nonzero. There is a finite decomposition
\[
\begin{gathered}
\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in R}\mathfrak g_\alpha,\\
\mathfrak g_\alpha=\{X:[T,X]=\alpha(T)X\text{ for all }T\in\mathfrak h\},
\qquad R\subset\mathfrak h^*\setminus\{0\}.
\end{gathered}
\tag{Q.5}
\]
The roots \(R\) span the complex dual \(\mathfrak h^*\). Put \(\mathfrak g_0=\mathfrak h\), and interpret a missing nonzero weight space as zero. Then
\[
[\mathfrak g_\alpha,\mathfrak g_\beta]\subseteq\mathfrak g_{\alpha+\beta}.
\tag{Q.6}
\]
The Killing form is nondegenerate on \(\mathfrak h\) and on every opposite pairing \(\mathfrak g_\alpha\times\mathfrak g_{-\alpha}\); all other weight-space pairings are zero.

**Proof.** [Symmetric spaces H.5–H.6](symmetric-spaces.md#theorem-h-5) give nondegeneracy of \(K\), zero centre, and commuting Jordan parts \(X=X_s+X_n\) inside \(\mathfrak g\). These parts have the adjoint Jordan operators. If \(Y\) commutes with \(X\), then \(\operatorname{ad}Y\) commutes with their polynomial expressions in \(\operatorname{ad}X\). Zero centre consequently gives \([Y,X_s]=[Y,X_n]=0\).

If every element had nilpotent adjoint operator, Engel's theorem, [Symmetric spaces H.4](symmetric-spaces.md#theorem-h-4), would make \(\mathfrak g\) nilpotent and hence solvable. A nonzero semisimple algebra cannot have this property. Thus when \(\mathfrak g\ne0\), some \(X_s\ne0\) supplies a one-dimensional toral subalgebra. Increase its dimension while possible; finite dimension gives a maximal one. More generally every given toral subalgebra lies in a maximal one by the same argument.

For any toral \(\mathfrak h\), a finite basis gives commuting diagonalizable adjoint operators. Simultaneous diagonalization from [Symmetric spaces H.1](symmetric-spaces.md#lemma-h-1) yields a joint weight decomposition, initially with zero space \(\mathfrak z=C_{\mathfrak g}(\mathfrak h)\). Jacobi gives (Q.6). Killing invariance gives
\[
(\alpha(T)+\beta(T))K(X,Y)=0
\quad(X\in\mathfrak g_\alpha,\ Y\in\mathfrak g_\beta).
\tag{Q.7}
\]
If the two weights do not add to zero, some \(T\) makes that coefficient nonzero. Nondegeneracy on the whole algebra then implies nondegeneracy on \(\mathfrak z\) and on each opposite pairing: a vector annihilating its only possible partner space would annihilate all of \(\mathfrak g\).

Suppose now that \(\mathfrak h\) is maximal toral. For \(X\in\mathfrak z\), the Jordan parts still lie in \(\mathfrak z\) by the first paragraph. Since \(X_s\) commutes with \(\mathfrak h\), adjoining it produces an abelian algebra of commuting diagonalizable operators, all of whose linear combinations are diagonalizable. Maximality therefore gives \(X_s\in\mathfrak h\). On \(\mathfrak z\) its adjoint is zero, so \(\operatorname{ad}X|_{\mathfrak z}=\operatorname{ad}X_n|_{\mathfrak z}\) is nilpotent. Engel's theorem makes \(\mathfrak z\) nilpotent, hence solvable.

Apply Lie's theorem, [Symmetric spaces H.3](symmetric-spaces.md#theorem-h-3), to the representation of this solvable algebra on \(\mathfrak g\). In a common triangular basis \(\operatorname{ad}X_n\) has zero diagonal, because it is nilpotent on the entire algebra. For every \(Y\in\mathfrak z\), the product \(\operatorname{ad}X_n\operatorname{ad}Y\) therefore has trace zero. Thus \(K(X_n,\mathfrak z)=0\). The restricted nondegeneracy already proved forces \(X_n=0\). Every \(X\in\mathfrak z\) consequently belongs to \(\mathfrak h\), proving \(\mathfrak z=\mathfrak h\) and (Q.5).

If \(T\in\mathfrak h\) is annihilated by all roots, it commutes with every summand of (Q.5), so is central and zero. The roots thus span \(\mathfrak h^*\): if their span were proper, solving the corresponding homogeneous linear equations would give a nonzero annihilator. For \(\mathfrak g\ne0\), the root set is nonempty, since otherwise \(\mathfrak g=\mathfrak h\) would be a nonzero abelian algebra. This proves the theorem. □

**Theorem Q.3 (coroot triples, root dimensions and reducedness).** Define \(T_\alpha\in\mathfrak h\) by
\[
K(T_\alpha,T)=\alpha(T)\quad(T\in\mathfrak h).
\tag{Q.8}
\]
Then \(q_\alpha=\alpha(T_\alpha)\ne0\). There are vectors \(e_\alpha\in\mathfrak g_\alpha\), \(f_\alpha\in\mathfrak g_{-\alpha}\) such that
\[
h_\alpha=\frac{2T_\alpha}{q_\alpha},\qquad
[h_\alpha,e_\alpha]=2e_\alpha,\quad
[h_\alpha,f_\alpha]=-2f_\alpha,\quad
[e_\alpha,f_\alpha]=h_\alpha.
\tag{Q.9}
\]
They span a copy of \(\mathfrak{sl}_2(\mathbb C)\). Every root space is one-dimensional. If \(c\alpha\in R\) for \(c\in\mathbb C\), then \(c=1\) or \(c=-1\). Moreover \(\beta(h_\alpha)\in\mathbb Z\) for all roots \(\alpha,\beta\).

**Proof.** Equation (Q.8) has a unique solution by Q.2. For \(e\in\mathfrak g_\alpha,f\in\mathfrak g_{-\alpha}\), Killing invariance gives, for every \(T\in\mathfrak h\),
\[
K([e,f],T)=K(e,[f,T])=\alpha(T)K(e,f).
\]
Since \([e,f]\in\mathfrak h\), nondegeneracy there yields
\[
[e,f]=K(e,f)T_\alpha.
\tag{Q.10}
\]
Choose \(e,f\) with \(K(e,f)=1\), using the opposite pairing of Q.2.

If \(q_\alpha=0\), then \(T_\alpha\) commutes with \(e,f\). Their span with \(T_\alpha\) is a solvable algebra, with central derived algebra. Lie's theorem makes its adjoint operators on \(\mathfrak g\) simultaneously upper triangular. In particular
\(\operatorname{ad}T_\alpha=[\operatorname{ad}e,\operatorname{ad}f]\)
is strictly triangular and nilpotent. But \(T_\alpha\in\mathfrak h\) has diagonalizable adjoint, so that adjoint is zero. Zero centre gives \(T_\alpha=0\), contrary to the nonzero functional \(\alpha\) in (Q.8). Thus \(q_\alpha\ne0\).

Keep \(e_\alpha=e\) and put \(f_\alpha=2f/q_\alpha\). Equations (Q.8)–(Q.10) give (Q.9). The three vectors are independent because they belong to the distinct weights \(\alpha,-\alpha,0\) of (Q.5), and their nonzero brackets are those of the three-by-basis algebra in Q.1. This identifies the subalgebra with \(\mathfrak{sl}_2\). Apply Q.1 to its action on \(\mathfrak g\): \(\beta(h_\alpha)\), an eigenvalue of \(\operatorname{ad}h_\alpha\) on \(\mathfrak g_\beta\), is an integer.

To prove the dimensions without assuming reducedness, use the invariant subspace
\[
M=\mathfrak h\oplus
\bigoplus_{\gamma\in R\cap\mathbb C\alpha}\mathfrak g_\gamma
\tag{Q.11}
\]
for this rank-one action. Its weight-zero space is exactly \(\mathfrak h\): if \(\gamma=c\alpha\), its \(h_\alpha\)-weight is \(2c\), which is zero only for \(c=0\). The map
\[
\operatorname{ad}e_\alpha:\mathfrak h\longrightarrow M,\qquad
T\longmapsto-\alpha(T)e_\alpha
\]
has rank one. In Q.1's string decomposition, every nontrivial even string has a one-dimensional zero-weight space which raises nontrivially. Odd strings have no zero weight and trivial strings contribute zero to this rank. Hence precisely one nontrivial even string occurs. The already embedded triple is \(V_2\), so it is that unique even string.

It follows that the weight-two space of \(M\), namely \(\mathfrak g_\alpha\), has dimension one, and the weight-four space is zero. Thus \(2\alpha\notin R\). This argument applies to every root.

Finally suppose \(\gamma=c\alpha\in R\). The integrality just proved gives \(2c=\gamma(h_\alpha)\in\mathbb Z\setminus\{0\}\), so \(c\) is real. By (Q.8), \(T_\gamma=cT_\alpha\) and \(q_\gamma=c^2q_\alpha\), hence \(h_\gamma=h_\alpha/c\). Integrality again gives \(2/c=\alpha(h_\gamma)\in\mathbb Z\setminus\{0\}\). After changing the sign of \(\gamma\), assume \(c>0\). The positive integers \(2c,2/c\) multiply to four; therefore \(c\in\{1/2,1,2\}\). The double-root exclusion rules out two, and its application to \(\gamma\) rules out one half. Only \(c=1\) remains. This proves reducedness even for complex proportionality. □

**Theorem Q.4 (complete root strings and nonzero brackets).** For \(\beta\ne\pm\alpha\), the roots on \(\beta+\mathbb Z\alpha\) form one uninterrupted string
\[
\beta-r\alpha,\ldots,\beta,\ldots,\beta+q\alpha,\qquad
r,q\in\mathbb Z_{\geq0},
\tag{Q.12}
\]
whose endpoints satisfy
\[
r-q=\beta(h_\alpha).
\tag{Q.13}
\]
Whenever \(\alpha+\beta\) is a root,
\[
[\mathfrak g_\alpha,\mathfrak g_\beta]=\mathfrak g_{\alpha+\beta}\ne0.
\tag{Q.14}
\]
The linear map on \(\mathfrak h^*\)
\[
s_\alpha\lambda=\lambda-\lambda(h_\alpha)\alpha
\tag{Q.15}
\]
permutes \(R\).

**Proof.** Q.3 makes \(\alpha,\beta\) nonproportional under the stated hypothesis. The sum
\[
M_{\alpha,\beta}=\bigoplus_{\beta+j\alpha\in R}\mathfrak g_{\beta+j\alpha}
\]
is invariant under the \(\alpha\) triple, by (Q.6). Its distinct \(h_\alpha\)-weights are \(\beta(h_\alpha)+2j\), each with multiplicity one by Q.3, and all are integers of the same parity.

By Q.1 this module is a sum of irreducible strings. Two even strings would both contribute weight zero, contradicting multiplicity one. Two odd strings would both contribute weight one, with the same contradiction. Since every constituent has the module's single parity, precisely one constituent occurs. Its weights are consecutive with step two and symmetric about zero. As the weight belonging to \(\beta\) occurs, the indices are the complete interval \(-r,\ldots,q\). Adding the two endpoint weights gives
\[
\bigl(\beta(h_\alpha)-2r\bigr)
+\bigl(\beta(h_\alpha)+2q\bigr)=0,
\]
which is (Q.13).

Within the irreducible string the raising operator is nonzero except at its upper endpoint, by (Q.3). Thus if \(\alpha+\beta\) exists, the bracket with \(e_\alpha\) maps the one-dimensional \(\beta\)-space nontrivially to the one-dimensional \(\alpha+\beta\)-space, proving (Q.14). Proportional roots cause no missing case in that assertion: reducedness excludes \(2\alpha\), and \(\alpha+(-\alpha)=0\) is not a root.

Negating a string weight sends the index \(j=0\) to the integer index \(-\beta(h_\alpha)\). Thus \(\beta-\beta(h_\alpha)\alpha\) is a root. For \(\beta=\pm\alpha\), (Q.15) exchanges \(\alpha\) and \(-\alpha\). These cases prove reflection closure. The map has square one because \(\alpha(h_\alpha)=2\), so it permutes the whole finite set. □

**Theorem Q.5 (the positive real Cartan space and its root system).** Let
\[
\mathfrak h_{\mathbb R}=\operatorname{span}_{\mathbb R}
\{h_\alpha:\alpha\in R\}.
\tag{Q.16}
\]
Then
\[
\mathfrak h=\mathfrak h_{\mathbb R}\oplus i\mathfrak h_{\mathbb R},
\]
and \(K|_{\mathfrak h_{\mathbb R}}\) is real and positive definite. The restrictions of \(R\) to the real dual \(E=\mathfrak h_{\mathbb R}^*\), with the inverse Killing inner product, form a finite reduced crystallographic root system. In that Euclidean product,
\[
h_\alpha=\frac{2T_\alpha}{(\alpha,\alpha)},\qquad
\beta(h_\alpha)=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}.
\tag{Q.17}
\]
Its simple bases and its finite-type diagram classification are those of P.1–P.3. For \(\mathfrak g=0\), all these spaces are zero and \(R\) is empty.

**Proof.** The roots span the complex dual by Q.2, so their Killing-dual vectors \(T_\alpha\), and consequently their nonzero scalar multiples \(h_\alpha\), span \(\mathfrak h\) over \(\mathbb C\). Every root takes real values on \(\mathfrak h_{\mathbb R}\), since its values on the spanning coroots are integers by Q.3. If \(X\in\mathfrak h_{\mathbb R}\cap i\mathfrak h_{\mathbb R}\), every root value on \(X\) is both real and purely imaginary, hence zero. Q.2 then gives \(X=0\). Together with complex spanning this proves the direct-sum assertion.

The simultaneous diagonalization in (Q.5), with root dimensions from Q.3, gives for \(X,Y\in\mathfrak h\)
\[
K(X,Y)=\sum_{\alpha\in R}\alpha(X)\alpha(Y).
\tag{Q.18}
\]
On \(\mathfrak h_{\mathbb R}\) all summands are real. At a nonzero \(X\), some root value is nonzero, again by Q.2, so \(K(X,X)\) is a strictly positive sum of real squares.

The real restrictions are distinct and nonzero: a complex-linear functional on \(\mathfrak h_{\mathbb R}\oplus i\mathfrak h_{\mathbb R}\) is determined by its restriction to the first summand. They span \(E\), because their real common annihilator is zero. For any \(\alpha\), solve its dual equation using the real positive Gram matrix of \(K|_{\mathfrak h_{\mathbb R}}\); this gives a real vector whose complexification satisfies (Q.8). Uniqueness in (Q.8) identifies it with \(T_\alpha\). Hence
\[
q_\alpha=\alpha(T_\alpha)=K(T_\alpha,T_\alpha)=(\alpha,\alpha)>0,
\]
and (Q.17) follows.

The set \(R\) is finite and spans \(E\); Q.3 gives reducedness and the integer Cartan numbers in (Q.17); Q.4 gives reflection closure. Formula (Q.15), together with (Q.17), is exactly the Euclidean orthogonal reflection (P.1). Thus all axioms of the root system in Part P have now been verified, and P.1–P.3 apply with their complete programme proofs.

If \(\mathfrak g\ne0\), Q.2 makes the Cartan subalgebra and the roots nonzero. For the zero algebra the stated choices verify every assertion directly. This positive real form is a real form of the Cartan vector space; it is not a compact real form of the whole Lie algebra. □

## R. Independence of the Cartan subalgebra

**Theorem R.1 (Cartan conjugacy and invariant Euclidean root data).** Any two Cartan subalgebras of a complex semisimple \(\mathfrak g\) are carried to one another by a finite product of automorphisms \(\exp(\operatorname{ad}X)\), \(X\in\mathfrak g\). Such an automorphism maps their canonical real Cartan spaces of Q.5 onto one another and identifies their root systems isometrically for the inverse Killing metrics. In particular the root data and diagram are independent of the Cartan choice.

**Proof.** The zero algebra is immediate, so suppose \(\mathfrak g\ne0\). We first give the elementary facts about complex polynomials used to connect the relevant elements.

For a nonzero polynomial \(p\) on \(\mathbb C^N\), the set \(U_p=\{p\ne0\}\) is open, dense and path connected. Continuity gives openness. A polynomial vanishing on a nonempty open set is zero: put a product of open discs inside that set, regard the polynomial as a polynomial in the first variable, and use that a nonzero one-variable polynomial has only finitely many roots. Its coefficient polynomials vanish on the remaining product; induction on \(N\) finishes the argument. The one-variable fact follows from division by a linear factor, proved in [Symmetric spaces H.1](symmetric-spaces.md#lemma-h-1). This proves density of \(U_p\).

For path connectedness, if \(x,y\in U_p\), the one-variable polynomial
\[
t\longmapsto p((1-t)x+ty)
\]
is not zero, since its values at zero and one are nonzero. Its zero set is finite. In the complex \(t\)-plane one can join zero to one while avoiding that set: start with the real segment and, around each zero on it, replace a small subsegment by a semicircle in a small disc containing no other zero and neither endpoint. The resulting continuous piecewise smooth path maps into \(U_p\). This proves all three facts.

Write the characteristic polynomial of \(\operatorname{ad}X\) as
\[
\det(tI-\operatorname{ad}X)=\sum_{j=0}^{N}c_j(X)t^j,\qquad
N=\dim_{\mathbb C}\mathfrak g.
\tag{R.1}
\]
Its coefficients are polynomial functions of the coordinates of \(X\), by the finite determinant expansion. Let \(r\) be the smallest multiplicity of the root zero among these polynomials as \(X\) varies. Thus \(c_j\) vanishes identically for \(j<r\), while \(c_r\) is not the zero polynomial. The set
\[
U=\{X:c_r(X)\ne0\}
\tag{R.2}
\]
consists exactly of the elements whose adjoint has zero eigenvalue with multiplicity \(r\), counting generalized eigenvectors. The preceding polynomial argument makes \(U\) nonempty, open, dense and connected.

Let \(\Gamma\) be the group of finite products of \(\exp(\operatorname{ad}X)\). These exponentials are automorphisms: for any derivation \(D\), differentiation gives
\[
\frac{d}{dt}\bigl(e^{-tD}[e^{tD}Y,e^{tD}Z]\bigr)=0,
\]
using the derivation rule, so the bracket is preserved at \(t=1\). Smoothness of the exponential and its derivative at zero are supplied by [Local tools 2.3](local-tools-for-bundles-and-transport.md#2-differential-equations-and-their-parameters), applied to the matrix group. Inverses are obtained by replacing \(X\) by \(-X\). Automorphisms conjugate adjoint operators and preserve every coefficient in (R.1), so \(\Gamma\) preserves \(U\).

Fix a Cartan \(\mathfrak h\) of dimension \(d\). Choose \(H\in\mathfrak h\) with \(\alpha(H)\ne0\) for every root. The finite-hyperplane argument in [De Rham decomposition AD.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ad-5) applies over \(\mathbb C\) as well: when varying a vector along \(tu\), each nonconstant affine root value forbids at most one scalar \(t\). Thus such choices exist. Put \(\mathfrak m=\bigoplus_{\alpha\in R}\mathfrak g_\alpha\), and consider
\[
\Phi:\mathfrak m\times\mathfrak h\longrightarrow\mathfrak g,\qquad
\Phi(Z,T)=\exp(\operatorname{ad}Z)T.
\tag{R.3}
\]
At \((0,H)\) its derivative is
\[
D\Phi_{(0,H)}(Z,T)=[Z,H]+T.
\]
On each root summand the first term is \(-\alpha(H)Z\), an invertible scalar, and on \(\mathfrak h\) the second term is the identity. Thus the derivative is an isomorphism. Apply the smooth real inverse function theorem, [Local tools 1.2](local-tools-for-bundles-and-transport.md#1-contraction-and-local-inversion), to the underlying real vector spaces. A small neighbourhood of \((0,H)\), restricted so that every root remains nonzero on \(T\), maps onto an open neighbourhood of \(H\).

Every element in that image is conjugate to such a \(T\); (Q.5) shows that its zero eigenvalue multiplicity is \(d\). This open image meets the dense set \(U\), forcing \(d=r\). The argument works for every Cartan, so all Cartan subalgebras have dimension \(r\).

Now let \(X\in U\). Its Jordan semisimple part \(X_s\) lies in some maximal toral \(\mathfrak h\): start with \(\mathbb C X_s\), which is toral even if it is zero, and use Q.2. The Jordan decomposition of \(\operatorname{ad}X\) shows that \(\operatorname{ad}X_s\) has the same eigenvalues with the same algebraic multiplicities. Its zero eigenspace therefore has dimension \(r\). It contains \(\mathfrak h\), whose dimension is \(r\), so
\[
C_{\mathfrak g}(X_s)=\mathfrak h.
\tag{R.4}
\]
The commuting nilpotent part \(X_n\) lies in this Cartan. Its adjoint is consequently both nilpotent and diagonalizable, hence zero. Zero centre gives \(X_n=0\). Thus every \(X\in U\) is semisimple and has Cartan centralizer \(C_{\mathfrak g}(X)=\mathfrak h\).

This Cartan is unique among those containing \(X\): any such Cartan lies in \(C_{\mathfrak g}(X)\) and has the same dimension \(r\). Conversely each Cartan contains an \(H\) as used in (R.3), for which the zero multiplicity is \(r\); hence every Cartan occurs as \(C_{\mathfrak g}(H)\) with \(H\in U\).

Partition \(U\) by \(\Gamma\)-conjugacy of these centralizers. Each class is open in \(U\). Indeed, around any \(H\in U\) with Cartan centralizer \(\mathfrak h\), the inverse-function neighbourhood (R.3) consists of elements conjugate to root-regular \(T\in\mathfrak h\), whose centralizer is exactly \(\mathfrak h\). Thus their centralizers are all conjugate to that of \(H\). If the partition had more than one class, one class and the union of all others would be disjoint nonempty open subsets separating \(U\). Its connectedness excludes this. There is one class, and since every Cartan occurs, they are all conjugate by \(\Gamma\).

Finally an automorphism \(F\) preserves \(K\), because
\(\operatorname{ad}(FX)=F(\operatorname{ad}X)F^{-1}\)
and trace is invariant under conjugation. If \(F\mathfrak h=\mathfrak h'\), it carries \(\mathfrak g_\alpha\) onto the root space for
\(\alpha'=\alpha\circ F^{-1}|_{\mathfrak h'}\).
Equation (Q.8) then gives \(F T_\alpha=T_{\alpha'}\), and (Q.9) gives \(Fh_\alpha=h_{\alpha'}\). Taking real spans proves \(F\mathfrak h_{\mathbb R}=\mathfrak h'_{\mathbb R}\). Its restriction preserves the real Killing inner product, so the induced dual map is an isometry carrying every root to its corresponding root. Part P now supplies the basis-independent diagram. □

## S. Words and the unreduced Lie presentation

In Parts S–U the field is \(\mathbb C\). Formal Lie algebras may initially be infinite dimensional; finiteness will be a conclusion. Fix an integral \(r\times r\) matrix \(A=(a_{ij})\) satisfying (O.1), including a positive diagonal \(D\) for which \(DA\) is symmetric positive definite. The empty matrix is allowed. We use the row convention
\[
\alpha_j(h_i)=a_{ij}.
\tag{S.1}
\]

**Lemma S.1 (formal free Lie algebras and ordered words).** A finite set \(X\) has a free Lie algebra \(F(X)\). It embeds in the free associative algebra \(T(X)\), with Lie bracket \(uv-vu\). More generally, if a complex Lie algebra \(L\) has a finite or countable ordered vector-space basis \((b_i)\), its enveloping algebra
\[
U(L)=T(L)/(xy-yx-[x,y]:x,y\in L)
\tag{S.2}
\]
has basis \(1\) and the words \(b_{i_1}\cdots b_{i_n}\) with \(i_1\leq\cdots\leq i_n\). In particular \(L\to U(L)\) is injective.

**Proof.** Take the vector space on all finite fully bracketed words in \(X\). Bracketing two words and extending bilinearly gives a nonassociative algebra. Quotient by the smallest subspace stable under bracketing on either side which contains antisymmetry and Jacobi expressions. The resulting bracket is a Lie bracket. Any map from \(X\) to a Lie algebra evaluates bracketed words, annihilates this subspace, and factors uniquely through the quotient. This constructs \(F(X)\) and its universal property. Its bracketed words form a countable spanning set; retaining a word precisely when it is not in the span of previously retained words gives a finite or countable basis.

For (S.2), \(T(L)\) is the vector space on ordinary words in basis letters, with concatenation as multiplication and the empty word as \(1\). Replace any adjacent descending pair by
\[
b_jb_i\ \longmapsto\ b_ib_j+[b_j,b_i]\qquad(j>i),
\tag{S.3}
\]
expanding the bracket in the basis. Order words first by length and then by the number of inverted pairs of indices. Each resulting term has smaller measure: the swapped term loses one inversion, and every bracket term has shorter length. Every expansion is finite; the length is bounded by the original length and the inversion count at each length is bounded by \(\binom n2\). Thus repeated reductions terminate.

We prove by induction on this measure that the resulting linear combination of ordered words is independent of all reduction choices. Two disjoint adjacent reductions commute after distributing sums, and the induction hypothesis applies to every resulting smaller word. The only overlapping choices have three letters \(c>b>a\). Reducing the three-letter part in the two orders gives, before reducing the shorter terms,
\[
\begin{aligned}
abc+[b,a]c+b[c,a]+[c,b]a,\\
abc+a[c,b]+[c,a]b+c[b,a].
\end{aligned}
\]
Their difference reduces to
\[
[[b,a],c]+[b,[c,a]]+[[c,b],a]=0.
\tag{S.4}
\]
Indeed the difference is the sum of three two-factor commutators; (S.3), bilinearity and antisymmetry reduce each to its bracket. With any words before and after these three letters, all these correction terms have strictly smaller total length than the original word. The induction hypothesis therefore justifies those reductions even in their surrounding context. Jacobi proves the last equality. These are all possible pairs of distinct first choices. If the first choice is the same, induction applies to its terms. This completes the induction and defines a unique linear normal-form map.

The map kills each expression \(p(xy-yx-[x,y])q\). For basis vectors this is one reduction, or its reverse when their indices increase, or zero for equal indices; uniqueness gives the same normal form in every context \(p,q\). Bilinearity gives the assertion for arbitrary \(x,y\). It therefore kills the entire defining two-sided ideal. Ordered words are fixed by the map and are linearly independent before quotienting. They consequently remain independent in the quotient; reductions show they span it. Length-one words prove that \(L\to U(L)\) is injective.

The commutator in any associative algebra is a Lie bracket: antisymmetry is immediate, and expanding Jacobi cancels the six ordered triple products in opposite pairs. An associative algebra map from \(U(L)\) is determined exactly by a linear map on \(L\) preserving brackets as commutators: evaluate tensor words, and (S.2) is precisely the condition for factorization. Apply this and the free-Lie universal property to \(F(X)\). There are maps
\[
T(X)\longrightarrow U(F(X))\longrightarrow T(X)
\]
fixing all letters of \(X\). Their composites are identities: \(T(X)\) is generated associatively by \(X\), and \(U(F(X))\) is generated by \(F(X)\), whose elements are Lie expressions in \(X\). Hence \(U(F(X))\cong T(X)\). The established injection of \(F(X)\) into its enveloping algebra gives the asserted free-Lie embedding. This proof also covers \(X=\varnothing\), for which \(F(X)=0\). □

**Lemma S.2 (independent Cartan generators and a free triangular decomposition).** Let \(\widetilde{\mathfrak g}_A\) be the Lie algebra on \(e_i,f_i,h_i\), subject only to
\[
\begin{gathered}
[h_i,h_j]=0,\qquad [h_i,e_j]=a_{ij}e_j,\\
[h_i,f_j]=-a_{ij}f_j,\qquad [e_i,f_j]=\delta_{ij}h_i.
\end{gathered}
\tag{S.5}
\]
Then the \(h_i\) are independent, and, as vector spaces,
\[
\widetilde{\mathfrak g}_A
=F(f_1,\ldots,f_r)\ \oplus\ \mathfrak h\
\oplus\ F(e_1,\ldots,e_r),
\qquad \mathfrak h=\bigoplus_i\mathbb C h_i.
\tag{S.6}
\]
All three summands embed by their indicated generators.

**Proof.** The presentation exists by S.1: take the free Lie algebra on the \(3r\) symbols and quotient by the ideal of (S.5). Give these symbols degrees \(-\varepsilon_i,0,\varepsilon_i\) for \(f_i,h_i,e_i\), respectively. The relations are homogeneous for this \(\mathbb Z^r\)-grading, which therefore descends to the quotient.

Write \(\widetilde{\mathfrak n}_\pm\) for the subalgebras generated by the corresponding signs. Jacobi and (S.5) show that \(\operatorname{ad}h_i\) preserves them. Induction on the length of a positive bracketed word gives
\[
[f_i,\widetilde{\mathfrak n}_+]\subset
\mathfrak h+\widetilde{\mathfrak n}_+.
\]
At length one this is (S.5); at a bracket of two shorter positive words apply Jacobi, and commute any resulting \(h_j\) past the remaining positive word. Interchanging signs gives the analogous inclusion for \([e_i,\widetilde{\mathfrak n}_-]\). Consequently the sum of the three indicated spaces is stable under the adjoint action of every generator and contains the generators. Repeated use of \([[x,y],z]=[x,[y,z]]-[y,[x,z]]\) expresses any bracketed word as a sum of iterated adjoint actions of generators on a generator. The three spaces thus span the presented algebra.

To show that they have no hidden relations, let \(T(f)\) be the free associative algebra on the \(f\)-letters and put
\[
\mathcal V=T(f)\otimes\mathbb C[t_1,\ldots,t_r].
\]
For a word \(w=f_{j_1}\cdots f_{j_s}\), set
\(\mu_i(w)=\sum_{\nu=1}^s a_{i j_\nu}\).
Define polynomial-linear operators, writing the tensor product as \(wP(t)\), by
\[
\begin{aligned}
F_i(wP)&=f_iwP,\\
H_i(wP)&=w(t_i-\mu_i(w))P,\\
E_i(1P)&=0,\\
E_i(f_jwP)&=f_jE_i(wP)+\delta_{ij}H_i(wP).
\end{aligned}
\tag{S.7}
\]
The last rule is a recursion on word length, so it defines \(E_i\) uniquely. The \(H_i\) commute, and the first two lines give \([H_i,F_j]=-a_{ij}F_j\). The recursion gives \([E_i,F_j]=\delta_{ij}H_i\). Every term of \(E_j(wP)\) is a polynomial times a word obtained by deleting one \(f_j\) from \(w\); its \(\mu_i\) is therefore \(\mu_i(w)-a_{ij}\). Computing \(H_iE_j-E_jH_i\) on each term gives \(a_{ij}E_j\). These are exactly all relations (S.5), so (S.7) defines a representation of \(\widetilde{\mathfrak g}_A\).

A Lie expression in the \(F_i\) acts by left multiplication by the corresponding commutator expression in \(T(f)\). A nonzero element of \(F(f)\) has a nonzero image there by S.1, and acting on the empty word detects it. Thus \(F(f)\to\widetilde{\mathfrak g}_A\) is injective. If \(\sum c_i h_i=0\), acting on \(1\) gives \(\sum c_i t_i=0\), so every \(c_i=0\).

The substitution
\[
\omega(e_i)=f_i,\qquad
\omega(f_i)=e_i,\qquad
\omega(h_i)=-h_i
\tag{S.8}
\]
preserves (S.5) and has square one. For example
\([f_i,e_j]=-\delta_{ij}h_i=\delta_{ij}\omega(h_i)\);
the \(h,e\) and \(h,f\) relations interchange with their required signs. Hence it is an automorphism, and it transfers the free injection on the negative side to the positive side. Positive words have degrees in \(\mathbb N^r\setminus\{0\}\), negative words have opposite degrees, and the \(h_i\) have degree zero. The grading makes their sum direct. This proves (S.6). □

## T. The Serre quotient and its finiteness

Set \(m_{ij}=1-a_{ij}\) for \(i\ne j\), and define
\[
S_{ij}^+=(\operatorname{ad}e_i)^{m_{ij}}e_j,\qquad
S_{ij}^-=(\operatorname{ad}f_i)^{m_{ij}}f_j.
\tag{T.1}
\]
Let \(\mathfrak g_A\) be the quotient of \(\widetilde{\mathfrak g}_A\) by all these elements.

**Lemma T.1 (the precise ideals on both sides).** If \(I_\pm\) is the ideal generated by the \(S_{ij}^\pm\) inside \(F(e)\) or \(F(f)\), respectively, then the ideal they generate in \(\widetilde{\mathfrak g}_A\) is exactly \(I_-\oplus I_+\). Consequently
\[
\mathfrak g_A=\mathfrak n_-\oplus\mathfrak h\oplus\mathfrak n_+,
\qquad
\mathfrak n_-=F(f)/I_-,\quad
\mathfrak n_+=F(e)/I_+.
\tag{T.2}
\]
The \(h_i\) remain independent and each \(e_i,f_i\) remains nonzero. Thus the relations (T.1) are precisely defining relations for the two indicated subalgebras.

**Proof.** We first compute in the unreduced algebra that
\[
[f_k,S_{ij}^+]=0\quad\hbox{for every }k.
\tag{T.3}
\]
For \(k\notin\{i,j\}\), \(f_k\) commutes with both letters. For \(k=i\), use \(E=\operatorname{ad}e_i\), \(F=\operatorname{ad}f_i\), \(H=\operatorname{ad}h_i\), and \(v=e_j\). They satisfy \([E,F]=H\), \([H,E]=2E\), \(Fv=0\), and \(Hv=a_{ij}v\). Expanding the commutator of \(F\) with \(E^m\) gives
\[
FE^m v
=-\sum_{\nu=0}^{m-1}E^\nu HE^{m-1-\nu}v
=-m(a_{ij}+m-1)E^{m-1}v.
\tag{T.4}
\]
Here \(HE^qv=(a_{ij}+2q)E^qv\), by induction from \([H,E]=2E\); summing the arithmetic progression gives the last equality. At \(m=m_{ij}\) its coefficient is zero. For \(k=j\), \(f_j\) commutes with \(e_i\), and hence
\[
[f_j,S_{ij}^+]=-(\operatorname{ad}e_i)^{m_{ij}}h_j.
\]
The first adjoint application gives \(-a_{ji}e_i\), and the second gives zero. If \(m_{ij}=1\), then \(a_{ij}=0\), so \(a_{ji}=0\) as well. This proves (T.3) in every case. The automorphism (S.8) gives \([e_k,S_{ij}^-]=0\).

Each \(S_{ij}^+\) is an \(h\)-weight vector. The positive ideal \(I_+\) is spanned by repeated adjoint actions of positive generators on these vectors: this span is stable under the generators, hence under all their bracketed words by Jacobi, and is the smallest such ideal. All those words are weight vectors, so \(I_+\) is \(h\)-stable. It is also stable under each \(\operatorname{ad}f_k\). Induct on the number of positive adjoint actions, starting with (T.3). In the step
\[
[f_k,[e_l,u]]
=[-\delta_{kl}h_l,u]+[e_l,[f_k,u]],
\]
both terms lie in \(I_+\), by weight stability and the induction hypothesis. Thus \(I_+\) is an ideal in the whole unreduced algebra. The same holds for \(I_-\) by (S.8).

Their sum is therefore an ideal containing all the relations (T.1), and every ideal containing those relations contains both \(I_+\) and \(I_-\). This proves equality with the full defining ideal. It is a direct sum by (S.6). Taking the quotient proves (T.2), including the exact positive and negative presentations. Each generator of \(I_+\) has positive total degree at least two; further positive brackets increase that degree. Thus \(I_+\) has no degree-\(\varepsilon_i\) part. The negative argument is identical. No part of either ideal has degree zero, so the \(h_i\) also survive independently.

Since \(A\) is invertible, the \(\alpha_j\) defined by (S.1) form a basis of \(\mathfrak h^*\). The multidegree \((c_j)\) corresponds to the weight \(\sum c_j\alpha_j\), because adjoint \(h_i\)-weights add under brackets. Distinct degrees therefore have distinct weights. Formula (T.2) says all nonzero weights have coefficients all nonnegative or all nonpositive, and the zero-weight space is exactly \(\mathfrak h\). Every fixed positive or negative degree has finite dimension: only finitely many bracketed words have that length and those letter multiplicities. Finally, a degree \(m\varepsilon_i\) with \(m\geq2\) is zero. Such words use just \(e_i\), or just \(f_i\), and every bracketed word in one letter of length at least two vanishes, by induction and \([e_i,e_i]=[f_i,f_i]=0\). □

**Lemma T.2 (reflection automorphisms before finiteness).** The derivations \(\operatorname{ad}e_i\) and \(\operatorname{ad}f_i\) are locally nilpotent on \(\mathfrak g_A\). The automorphism
\[
w_i=\exp(\operatorname{ad}e_i)
      \exp(-\operatorname{ad}f_i)
      \exp(\operatorname{ad}e_i)
\tag{T.5}
\]
satisfies
\[
w_i(e_i)=-f_i,\quad w_i(f_i)=-e_i,\quad
w_i(H)=H-\alpha_i(H)h_i\quad(H\in\mathfrak h).
\tag{T.6}
\]
It maps the weight space of \(\gamma\) isomorphically onto that of
\(\gamma-\gamma(h_i)\alpha_i\).

**Proof.** Write \(E=\operatorname{ad}e_i\). It kills \(e_i\), and (T.1) gives a vanishing power on each other \(e_j\). It sends \(h_j\) to \(-a_{ji}e_i\) and then to zero. It kills \(f_j\) for \(j\ne i\), while
\[
f_i\ \xmapsto{E}\ h_i\
\xmapsto{E}\ -2e_i\ \xmapsto{E}\ 0.
\]
It is consequently nilpotent on each generator. For any derivation \(N\), induction on \(q\) gives
\[
N^q[u,v]=\sum_{a=0}^q\binom qa[N^a u,N^{q-a}v].
\tag{T.7}
\]
If \(N^p u=N^s v=0\), every term vanishes for \(q\geq p+s-1\). Induction on bracketed words and then finite linear sums proves that \(E\) is locally nilpotent everywhere. The same argument, or (S.8), handles \(\operatorname{ad}f_i\).

For a locally nilpotent derivation \(N\), the exponential is a finite sum on each vector. Formula (T.7) proves \(\exp N[u,v]=[\exp Nu,\exp Nv]\); only finitely many terms occur for fixed \(u,v\). Multiplication of the finite power series on each such vector gives inverse \(\exp(-N)\). This proves that (T.5) is an automorphism without assuming finite dimensionality.

Put \(e=e_i,f=f_i,h=h_i\) and \(F=\operatorname{ad}f\). Directly from the three brackets \([h,e]=2e,[h,f]=-2f,[e,f]=h\),
\[
\begin{gathered}
\exp E(f)=f+h-e,\qquad \exp E(h)=h-2e,\\
\exp(-F)(e)=e+h-f,\qquad
\exp(-F)(h)=h-2f.
\end{gathered}
\]
For \(e\), the three factors in (T.5), acting from right to left, give \(e\), then \(e+h-f\), then \(-f\). For \(f\) they give \(f+h-e\), then \(-e\), then \(-e\). For a general \(H\in\mathfrak h\), write \(a=\alpha_i(H)\). The first factor to act sends it to \(H-ae\). Since \(\exp(-F)H=H-af\), the next sends this to \(H-ae-ah\), and the last sends it to \(H-ah\). This is (T.6).

The restriction of \(w_i\) to \(\mathfrak h\) is an involution, since \(\alpha_i(h_i)=2\). For a weight vector \(x\), compute
\[
[H,w_i x]=w_i[w_i^{-1}H,x]
=\bigl(\gamma(H)-\gamma(h_i)\alpha_i(H)\bigr)w_i x.
\]
The inverse gives equality of the indicated weight spaces, not just inclusion. Under the root realization P.1, \(\gamma(h_i)=2(\gamma,\alpha_i)/(\alpha_i,\alpha_i)\), so this is the simple-root reflection \(s_i\). □

**Theorem T.3 (exact finite root decomposition).** Let \(R_A\) be the finite root system of P.1 with ordered simple basis \(\alpha_1,\ldots,\alpha_r\) and Cartan matrix \(A\). Then
\[
\mathfrak g_A=\mathfrak h\oplus
\bigoplus_{\gamma\in R_A}(\mathfrak g_A)_\gamma,
\qquad \dim(\mathfrak g_A)_\gamma=1,
\qquad \dim\mathfrak g_A=r+|R_A|.
\tag{T.8}
\]
The subalgebra \(\mathfrak h\) is maximal toral and self-centralizing, and its row Cartan matrix is \(A\).

**Proof.** By T.1 the possible nonzero weights lie in the positive or negative integral cone on the \(\alpha_i\). Identify their real span with the Euclidean realization P.1, whose Gram matrix is \(DA\). Suppose \(\gamma=\sum c_i\alpha_i\) is a positive weight with nonzero weight space. If just one coefficient is positive, T.1 forces \(\gamma=\alpha_i\), since every higher pure multiple has zero space.

Otherwise at least two coefficients are positive. Since
\[
0<(\gamma,\gamma)=\sum_i c_i(\gamma,\alpha_i),
\]
there is an \(i\) with \(c_i>0\) and \((\gamma,\alpha_i)>0\). The number
\[
k=\gamma(h_i)=\sum_j c_j a_{ij}
=\frac{2(\gamma,\alpha_i)}{(\alpha_i,\alpha_i)}
\]
is a positive integer. T.2 takes its nonzero space to the space of \(s_i\gamma=\gamma-k\alpha_i\). At least one other positive coefficient remains. Thus \(s_i\gamma\) cannot lie in the negative cone or be zero. The one-sign restriction of T.1 forces it into the positive cone; its height \(\sum c_i\) has decreased by \(k\). Repeating reaches a simple root after finitely many steps.

It follows that every positive weight lies in the simple-reflection orbit of a simple root. The negative argument follows by changing all signs, using the same descent on \(-\gamma\). P.1 proves that the union of these orbits is precisely the finite set \(R_A\). Conversely each simple-root vector \(e_i\) is nonzero by T.1, and every word of simple reflections is realized on weights by a word of the automorphisms \(w_i\). Hence every member of \(R_A\) actually occurs. This argument does not require relations among the \(w_i\): a chosen word of root reflections is enough.

The simple-root spaces are one dimensional, spanned by their single degree-one generators. Descent and the isomorphisms in T.2 therefore show that every root space is one dimensional. The zero space is exactly \(\mathfrak h\), so (T.8) follows. In particular the algebra is finite dimensional, now as a proved fact.

Its weight decomposition makes every \(\operatorname{ad}H\), \(H\in\mathfrak h\), diagonalizable, and the zero-space statement says \(C_{\mathfrak g_A}(\mathfrak h)=\mathfrak h\). Thus \(\mathfrak h\) is toral and cannot be contained in a larger toral subalgebra. The surviving triples \(e_i,f_i,h_i\) and (S.1) give the asserted row Cartan matrix. For \(r=0\), S.1 gives the zero algebra and all statements have their empty meanings. □

**Theorem T.4 (simplicity and the component decomposition).** If the diagram of \(A\) is nonempty and connected, \(\mathfrak g_A\) is simple. For arbitrary \(A\), it is the direct sum of the simple algebras belonging to the connected components, and is semisimple.

**Proof.** First suppose the diagram is connected. Let \(J\ne0\) be an ideal. Choose \(H\in\mathfrak h\) on which the finitely many distinct weights \(R_A\cup\{0\}\) take distinct values. Such \(H\) exists: the product of their nonzero pairwise differences is a nonzero polynomial, which cannot vanish everywhere, as proved in R.1. The projection onto any eigenspace of \(\operatorname{ad}H\) is the polynomial
\[
\prod_{\lambda\ne\mu}
\frac{\operatorname{ad}H-\lambda I}{\mu-\lambda},
\tag{T.9}
\]
where \(\lambda,\mu\) run over its distinct eigenvalues. The ideal is stable under these polynomials and hence is the sum of its weight intersections.

It therefore contains either a nonzero root vector or a nonzero \(Z\in\mathfrak h\). In the latter case some \(\alpha_i(Z)\ne0\), since the simple roots are a basis, and \([Z,e_i]=\alpha_i(Z)e_i\) supplies a root vector. Every exponential in (T.5) preserves \(J\), as does its inverse: they are finite sums of adjoint operators preserving the ideal. The descent in T.3 therefore carries a root vector in \(J\) to a nonzero multiple of a simple \(e_i\) or \(f_i\). In the latter case \(w_i\) sends it to a multiple of \(e_i\). The root-space dimensions show \(e_i\in J\).

Now \(h_i=[e_i,f_i]\in J\), and \([h_i,f_i]=-2f_i\) gives \(f_i\in J\). If vertex \(j\) is adjacent to \(i\), then \(a_{ij}\ne0\), so \([h_i,e_j]=a_{ij}e_j\) gives \(e_j\in J\), and then also \(h_j,f_j\in J\). Connectivity propagates this to every generator. Thus \(J=\mathfrak g_A\). The algebra is nonabelian because \(h_i\ne0\), so it is simple.

For different components all generator brackets vanish across the components. This is already (S.5) for mixed signs or an \(h\); for equal signs, \(a_{ij}=0\) makes the Serre relation precisely \([e_i,e_j]=0\) or \([f_i,f_j]=0\). Jacobi then proves that the generated component subalgebras commute. The presentation for each diagonal block maps into \(\mathfrak g_A\). Sending all generators outside that block to zero respects every defining relation and is a left inverse, so the map is injective. More explicitly, the map from \(\mathfrak g_A\) to the direct sum of the block presentations, defined on its generators, respects the same relations; its inverse is the sum of those commuting inclusions. Both composites fix all generators. This proves the asserted direct sum.

Each nonzero simple summand is perfect, because its derived algebra is a nonzero ideal; it is consequently not solvable. The image of a solvable ideal under projection to any summand is a solvable ideal there, hence zero. Thus the direct sum has no nonzero solvable ideal and is semisimple. The empty sum is the zero semisimple algebra. Applying Q.5 now also gives its canonical positive real Cartan form. The Euclidean realization with Gram matrix \(DA\) is equivalent to that root system by P.1, with the permitted positive scale on each component; no prescribed \(D\) was asserted to be the Killing normalization. □

## U. Reconstruction and classification of complex semisimple algebras

**Theorem U.1 (existence, reconstruction and uniqueness).** Every matrix \(A\) satisfying (O.1) produces the finite-dimensional complex semisimple algebra \(\mathfrak g_A\) with presentation (S.5), (T.1). It has the exact triangular decomposition and defining positive and negative relations in T.1. Every finite-dimensional complex semisimple Lie algebra is isomorphic to one of these algebras. Two are isomorphic exactly when their oriented Cartan diagrams agree up to permutation of vertices and components. Consequently the isomorphism classes are classified by finite multisets of the connected diagrams in (P.4), including the empty multiset for the zero algebra.

**Proof.** Existence, the exact presentation, its noncollapse and finiteness are S.1–T.3, and semisimplicity is T.4. These proofs apply to every positive Cartan matrix, including the exceptional types in O.2.

Conversely let \(\mathfrak g\) be a finite-dimensional complex semisimple algebra. If it is zero, use the empty presentation. Otherwise choose a maximal toral \(\mathfrak h\) by Q.2, its root system by Q.3–Q.5, and a simple basis \(\alpha_1,\ldots,\alpha_r\) by P.1. Choose the actual triples \(e_i,f_i,h_i\) of Q.3, normalized so that
\[
[e_i,f_i]=h_i,\qquad
h_i=\frac{2T_{\alpha_i}}{K(T_{\alpha_i},T_{\alpha_i})},
\qquad
[h_i,e_i]=2e_i.
\tag{U.1}
\]
The \(h_i\) form a basis of \(\mathfrak h\), since the simple roots form a basis of its dual and the nondegenerate Killing duality takes them to the independent \(T_{\alpha_i}\). Set \(a_{ij}=\alpha_j(h_i)\). Q.5 and O.1 show that \(A\) satisfies all the required integral, sign and positive-symmetrization conditions.

The \(h,h\), \(h,e\) and \(h,f\) relations are the weight equations. For \(i\ne j\), \(\alpha_i-\alpha_j\) is not a root: its simple coefficients have both signs. The bracket of \(e_i,f_j\) is therefore zero by Q.2. For the positive Serre relation, the root string of \(\alpha_j\) in the \(\alpha_i\) direction cannot extend to \(\alpha_j-\alpha_i\); its lower endpoint parameter is \(0\). Q.4 gives the upper parameter \(-a_{ij}\), so
\[
(\operatorname{ad}e_i)^{1-a_{ij}}e_j=0.
\]
For the negative relation apply the same string theorem to \(-\alpha_j\) in the direction \(-\alpha_i\). Its Cartan integer is again \(a_{ij}\), giving the stated relation in the \(f_i\). All relations (S.5), (T.1) thus hold in \(\mathfrak g\), and the universal property gives a homomorphism \(\mathfrak g_A\to\mathfrak g\).

It is surjective. For a positive nonsimple root \(\beta=\sum c_i\alpha_i\), positivity of
\((\beta,\beta)=\sum c_i(\beta,\alpha_i)\)
gives an \(i\) with \(c_i>0\) and \((\beta,\alpha_i)>0\). The string formula Q.4 then has lower parameter strictly positive, so \(\beta-\alpha_i\) is a root. Its coefficients are nonnegative, it is nonzero, and its height is smaller. Induction on height, starting from the \(e_i\), and the nonzero root brackets in Q.4 show that the generated algebra contains \(\mathfrak g_\beta\). These spaces have dimension one by Q.3. The same argument for negative roots starts from the \(f_i\). Finally the brackets \([e_i,f_i]\) span \(\mathfrak h\), proving surjectivity.

The simple roots have Cartan matrix \(A\). P.1 identifies their entire root set with \(R_A\); Q.2–Q.3 give
\[
\dim\mathfrak g=r+|R_A|=\dim\mathfrak g_A.
\]
A surjective linear map between these finite-dimensional spaces has zero kernel, so the homomorphism is an isomorphism.

A permutation of a Cartan diagram gives an isomorphism of presentations by relabeling \(e_i,f_i,h_i\). Conversely a Lie-algebra isomorphism preserves the Killing form, since it conjugates each adjoint operator. It takes a maximal toral subalgebra to one of the target and transports its roots and their Cartan ratios. R.1 conjugates that target Cartan to the chosen one by an automorphism. P.2 compares the transported simple basis with any chosen simple basis there. Hence the Cartan matrices, and their oriented diagrams, agree up to simultaneous permutation of indices. This also permits permutation of the connected components. O.2 and P.3 give the finite diagram list and its exact rank conventions, so no further identifications occur. □

The classification just proved is for complex semisimple Lie algebras. Real forms and irreducible Hermitian symmetric pairs require further structure.

## V. Calculations with positive Cartan matrices

**Exercise V.1 (local obstructions and determinant tests).** For the normalized matrix \(Q\) of O.1, verify the bound on bond multiplicity and the obstructions from a vertex with four simple neighbours, two separated double bonds, or an edge adjoining a triple bond. Explain why the full determinant's sign alone is insufficient.

**Solution.** A two-vertex restriction with bond product \(k\) has diagonal two and off-diagonal \(-\sqrt k\). Its determinant is \(4-k\), and
\[
2x^2-2\sqrt k\,xy+2y^2
=2\left(x-\frac{\sqrt k}{2}y\right)^2
+\left(2-\frac{k}{2}\right)y^2.
\tag{V.1}
\]
It is positive definite exactly when \(k<4\). A bond has positive integral product, leaving \(1,2,3\). At \(k=4\), \((1,1)\) is null; at \(k>4\), its value \(4-2\sqrt k\) is negative.

On a simple four-leaf star take central coordinate \(2\) and leaf coordinates \(1\). The diagonal contribution is \(2(4+4)=16\), and the four edges contribute \(4(-2\cdot2\cdot1)=-16\). Extra supported edges or heavier bonds decrease this value; additional vertices can be assigned zero. Thus the same witness excludes every extension of this configuration.

For a five-vertex path with double end edges, take \((1,\sqrt2,\sqrt2,\sqrt2,1)\). Its diagonal contribution is \(16\), and each of its four edges contributes \(-4\). More generally, a path with \(m\geq3\) vertices, double end edges and simple intervening edges has the analogous witness with endpoint values one and all interior values \(\sqrt2\). Its quadratic value is
\[
(4m-4)-8-4(m-3)=0.
\]
Extra weights decrease it. Arrows do not alter this calculation: (O.3) depends only on the products.

A triple edge alone has determinant \(1\). Adding one simple edge at its endpoint gives the completed-square star coefficient \(2-(3+1)/2=0\) from (O.5). The corresponding three-vertex restriction is already nonpositive on a nonzero vector, excluding every larger connected extension.

Finally, \(\operatorname{diag}(1,-1,-1)\) has determinant \(1\) but takes negative values. Among the normalized graph matrices, a three-cycle of simple edges annihilates \((1,1,1)\). The arguments in O.2 use positive blocks and their completed-square remainders, which test the quadratic form; a final determinant sign would not supply those block hypotheses. □

**Exercise V.2 (sharp arm and internal-bond boundaries).** Compute determinants and definiteness at arm triples \((1,2,4)\), \((1,2,5)\), \((1,2,6)\), and \((1,1,n-3)\). For a path cut by a double edge into simple chains of sizes \(p,q\), examine \((2,2),(2,3),(2,4)\).

**Solution.** Give the empty chain determinant one. Endpoint expansion of the simple-chain matrix \(T_l\) gives
\(\det T_l=2\det T_{l-1}-\det T_{l-2}\).
The initial values \(1,2\) imply \(\det T_l=l+1\) by induction. Positivity of these blocks is (O.6).

The square completion (O.7) is a triangular change of variables of determinant one. It splits off \(B\), so the original determinant is \(\det(B)(d-b^tB^{-1}b)\). For three arms,
\[
c=\frac1{p+1}+\frac1{q+1}+\frac1{r+1}-1,\qquad
\det Q=(p+1)(q+1)(r+1)c.
\tag{V.2}
\]
The arm blocks remain positive, while the sign of \(c\) determines the remaining direction. Thus
\[
\begin{array}{c|c|c|c}
(p,q,r)&c&\det Q&\text{form}\\ \hline
(1,2,4)&1/30&1&\text{positive}\\
(1,2,5)&0&0&\text{singular}\\
(1,2,6)&-1/42&-1&\text{indefinite}.
\end{array}
\]
In the last case the arm directions are positive and the remaining direction negative, proving indefiniteness. For \((1,1,n-3)\), \(c=1/(n-2)\) and the determinant is \(2\cdot2\cdot(n-2)/(n-2)=4\), for every \(n\geq4\). At \(n=4\) the central coefficient is \(1/2\).

For the double-edge path, eliminate \(T_p\) to obtain \(T_q-\frac{2p}{p+1}e_1e_1^t\). Multilinearity in columns proves \(\det(I+uv^t)=1+v^tu\): terms using two updated columns vanish because those columns are proportional; the single-column terms sum to \(v^tu\). Factoring an invertible \(B\) therefore gives
\(\det(B-tvv^t)=\det B(1-tv^tB^{-1}v)\).
Using the inverse entry from (O.6) yields
\[
\det Q=(p+1)(q+1)-2pq.
\tag{V.3}
\]
The positivity test (O.9) is \(1-2pq/((p+1)(q+1))>0\). For the three requested pairs this expression is respectively \(1/9,0,-1/15\), and the determinants are \(1,0,-1\). These give the positive \(F_4\) case, its singular boundary, and an indefinite next case, since positive block directions remain in the last case. □

**Exercise V.3 (row conventions and classification scope).** Write \(B_3,C_3\) Cartan matrices and positive symmetrizers, explain why no vertex permutation identifies them, and state what O.2 classifies.

**Solution.** Number vertices along the chain with its double edge last. Then
\[
\begin{aligned}
A_B&=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix},
&D_B&=\operatorname{diag}(2,2,1),\\
A_C&=\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix},
&D_C&=\operatorname{diag}(1,1,2).
\end{aligned}
\tag{V.4}
\]
Multiplication verifies symmetry of \(D_BA_B,D_CA_C\); positivity follows from (O.9) with sizes \(2,1\). The squared lengths are \(2d_i\), so the arrow points toward vertex \(3\) in \(B_3\) and toward vertex \(2\) in \(C_3\). The endpoint belonging to the double edge is intrinsic to the valued graph. Any isomorphism must preserve that endpoint and whether it is short; the two diagrams differ. In rank two, exchanging both vertices identifies the names \(B_2,C_2\).

O.2 classifies integral matrices with diagonal two, nonpositive off-diagonal entries, paired zeros and a positive diagonal symmetrization, up to simultaneous index permutation. Its connected table, or a multiset of connected blocks, determines exact entries. O.1 determines relative lengths within each component. Root systems are constructed in P.1–P.3 and Lie brackets in S.1–U.1; a diagram's name supplies neither construction without those proofs. Real forms and Hermitian pairs require further structure. □

## W. Concrete rank-two roots and lattice checks

**Exercise W.1 (six, eight and twelve roots).** In the row convention realize
\[
A_m=\begin{pmatrix}2&-1\\-m&2\end{pmatrix},\qquad
S_m=\begin{pmatrix}2m&-m\\-m&2\end{pmatrix},
\qquad m=1,2,3,
\tag{W.1}
\]
by explicit roots in a basis \(a,b\) with Gram matrix \(S_m\). Verify the root axioms and the arrow direction.

**Solution.** The first principal entry is positive and the completed-square remainder is positive because \(\det S_m=m(4-m)>0\). The simple reflections on coefficients are
\[
s_a(x,y)=(-x+y,y),\qquad s_b(x,y)=(x,mx-y).
\tag{W.2}
\]
Take each following set together with its negatives:
\[
\begin{array}{c|l}
m&R_m^+\\ \hline
1&a,\ b,\ a+b\\
2&a,\ b,\ a+b,\ a+2b\\
3&a,\ b,\ a+b,\ a+2b,\ a+3b,\ 2a+3b.
\end{array}
\tag{W.3}
\]
The complete stability check on positive members is below; linearity handles their negatives.

| \(m\) | root | \(s_a\) image | \(s_b\) image |
|---|---|---|---|
| 1 | \(a\) | \(-a\) | \(a+b\) |
| 1 | \(b\) | \(a+b\) | \(-b\) |
| 1 | \(a+b\) | \(b\) | \(a\) |
| 2 | \(a\) | \(-a\) | \(a+2b\) |
| 2 | \(b\) | \(a+b\) | \(-b\) |
| 2 | \(a+b\) | \(b\) | \(a+b\) |
| 2 | \(a+2b\) | \(a+2b\) | \(a\) |
| 3 | \(a\) | \(-a\) | \(a+3b\) |
| 3 | \(b\) | \(a+b\) | \(-b\) |
| 3 | \(a+b\) | \(b\) | \(a+2b\) |
| 3 | \(a+2b\) | \(a+2b\) | \(a+b\) |
| 3 | \(a+3b\) | \(2a+3b\) | \(a\) |
| 3 | \(2a+3b\) | \(a+3b\) | \(2a+3b\) |

Every listed root is in the orbit of \(a\) or \(b\): \(a+b=s_ab\); for \(m=2\), \(a+2b=s_ba\); for \(m=3\), \(a+3b=s_ba\), \(a+2b=s_bs_ab\), and \(2a+3b=s_as_ba\). Negating a simple root is its own simple reflection, so the negative roots belong to these orbits too.

A product \(w\) of simple reflections is an isometry permuting the set. Reflection in \(wa\) or \(wb\) is \(ws_aw^{-1}\) or \(ws_bw^{-1}\), directly from (P.1). Hence reflection in every root preserves the set. If \(\alpha=wa\), then
\[
\frac{2(\beta,\alpha)}{(\alpha,\alpha)}
=\frac{2(w^{-1}\beta,a)}{(a,a)}
\]
is integral: \(w^{-1}\beta\) has integer coefficients and row one of \(A_m\) is integral. Row two handles the orbit of \(b\). The sets span because they contain \(a,b\). Inspection of (W.3) shows that no two distinct listed members are proportional, proving reducedness after adding negatives. The one-sign expansions make \(a,b\) a simple basis by P.1.

There are \(6,8,12\) roots, of types \(A_2,B_2,G_2\). In \(A_2\) every listed root has squared length two. In \(B_2\), \(a,a+2b\) have squared length four, and \(b,a+b\) squared length two. For \(m>1\), the arrow points toward \(b\), the shorter basis vector. In \(G_2\), substitution in \(S_3\) gives squared lengths \(2\) for \(b,a+b,a+2b\), and \(6\) for \(a,a+3b,2a+3b\). □

**Exercise W.2 (integrality, primitive vectors and positivity).** Explain why even lattice norms alone do not prove crystallographic integrality for long roots, why integral reflection orbits of basis vectors are primitive and reduced, and how height descent acts on the highest \(B_2\) root. Identify the failure at off-diagonal entries \(-2,-2\).

**Solution.** In the integral lattice with Gram matrix
\[
\begin{pmatrix}4&-1\\-1&2\end{pmatrix},
\]
every norm \(4x^2-2xy+2y^2\) is even for integral \(x,y\). The form is positive, by completing a square and using determinant \(7\). Yet for its basis vectors \(u,v\), \(2(u,v)/(u,u)=-1/2\). Norm two would remove that denominator; longer vectors require further divisibility. In W.1 it comes from the integral Cartan row and remains true under the integral isometries (W.2).

An integral invertible matrix with integral inverse preserves primitive vectors, meaning those whose coordinates have greatest common divisor one. If \(d>1\) divided all transformed coordinates, the integral inverse would make it divide all original coordinates. Every simple reflection is its own integral inverse, and every basis vector is primitive.

If primitive integral vectors satisfy \(y=cx\), a nonzero coordinate makes \(c=p/q\) rational, with coprime integers and \(q>0\). Here is the integer cancellation needed for this step. Among the positive integer combinations \(ap+bq\), with \(a,b\in\mathbb Z\), choose the least, \(d\); the set is nonempty since it contains \(q>0\). Divide \(p\) and \(q\) by \(d\) with remainder, choosing the integer quotients by the floor of their real ratios. Each nonzero remainder would be a smaller positive integer combination, so both remainders are zero. Hence \(d\) divides both \(p,q\); their coprimeness makes \(d=1\). Thus some \(a,b\) satisfy \(ap+bq=1\). For each coordinate, integrality of \(y_j=px_j/q\) gives \(q\mid px_j\), so \(q\) divides both terms of \(apx_j+bqx_j=(ap+bq)x_j=x_j\). Primitiveness of \(x\) now forces \(q=1\). Then primitiveness of \(y\) gives \(|p|=1\). Thus \(y=\pm x\), proving reducedness of any set made from such orbits. This does not assert that every primitive lattice vector is an orbit root. Finiteness alone would be insufficient: \(\{\pm a,\pm2a\}\) is finite and nonreduced.

In \(B_2\), \(\gamma=a+2b\) has coroot pairing \(-2+4=2\) with \(b^\vee\), so
\[
s_b\gamma=\gamma-2b=a.
\tag{W.4}
\]
Its height drops from three to one. The other nonsimple positive root \(a+b\) has pairing one with \(a^\vee\), and \(s_a(a+b)=b\), lowering height from two to one. In both cases a positive coefficient outside the reflected position remains, preventing the reflected root from being negative. The general positive-norm and one-sign arguments are T.3.

For the matrix
\[
\begin{pmatrix}2&-2\\-2&2\end{pmatrix},
\tag{W.5}
\]
the candidate Gram form has null vector \((1,1)\). Its formal coefficient reflections are \(s_1(x,y)=(-x+2y,y)\), \(s_2(x,y)=(x,2x-y)\), with
\[
s_1s_2=I+N,\qquad
N=\begin{pmatrix}2&-2\\2&-2\end{pmatrix},\qquad N^2=0.
\]
Induction gives \((I+N)^n=I+nN\) for \(n\geq0\), so the orbit of \((1,0)\) contains every \((1+2n,2n)\), an infinite set. The null vector \((1,1)\) is fixed by both reflections. Positive squared norm can no longer force a positive root pairing. Also invertibility no longer identifies different degrees with different Cartan weights as in T.1. These failures explain the positivity hypothesis; none of the finite-type conclusions applies to (W.5). □

**Exercise W.3 (long-root sublattices and mixed signs).** Find the stronger norm divisibility for long-root orbits in \(B_2,G_2\). In the coordinate model of \(C_4\), show that a mixed-sign vector can have long-root norm while still failing to be a root, and explain the additional invariant that excludes it.

**Solution.** In W.1 with \(m=2\) or \(3\), let
\[
M_m=\{(x,y)\in\mathbb Z^2:y\in m\mathbb Z\}.
\]
Both transformations (W.2) preserve it: \(s_a\) keeps \(y\), and \(s_b\) replaces it by \(mx-y\). It contains \(a=(1,0)\), so it contains the long-root orbit. On writing \(y=mz\), its squared norm is
\[
2m x^2-2m xy+2y^2
=2m(x^2-mxz+mz^2).
\tag{W.6}
\]
Positive norms on this sublattice are therefore at least \(2m\), not merely two. The long roots in W.1 have precisely norm \(2m\).

Here is how this stronger bound excludes mixed signs. For any coefficient vector \(c\), write \(c=p-q\), where \(p,q\) have nonnegative integer coordinates with disjoint supports. The off-diagonal Gram entries are nonpositive, so \((p,q)\leq0\). Consequently
\[
\|p-q\|^2\geq\|p\|^2+\|q\|^2.
\tag{W.7}
\]
If \(c\in M_m\), both \(p,q\) are in \(M_m\). If both are nonzero, (W.6) and positivity give norm at least \(4m\), exceeding a long root's \(2m\). For the ordinary integer lattice, norms in all three matrices (W.1) are positive even integers; (W.7) gives the weaker bound four. That suffices for every norm-two root, including all \(A_2\) roots, but cannot by itself exclude all long-root norms.

For the precise \(C_4\) example use the complete coordinate construction in [De Rham decomposition AE.3](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-3):
\[
\alpha_i=e_i-e_{i+1}\ (1\leq i\leq3),\qquad
\alpha_4=2e_4,
\]
with orthonormal \(e_i\). Its Gram matrix has diagonal \(2,2,2,4\) and adjacent entries \(-1,-1,-2\). The vector
\[
x=\alpha_1-\alpha_3=e_1-e_2-e_3+e_4
\]
has squared norm four and mixed simple coefficients. It is not a long root, since the actual long roots in the proved coordinate model are \(\pm2e_i\).

The relevant sublattice in simple coefficients is
\[
M=\{(c_1,c_2,c_3,c_4)\in\mathbb Z^4:
c_1,c_2,c_3\text{ are even}\}.
\tag{W.8}
\]
Indeed the orthonormal coordinates of \(\sum c_i\alpha_i\) are
\[
(c_1,\ -c_1+c_2,\ -c_2+c_3,\ -c_3+2c_4).
\]
They are all even exactly when \(c\in M\): one implication is immediate, and the reverse follows by successive addition of the first three coordinates. Thus \(M\) is exactly \(2\mathbb Z^4\) in orthonormal coordinates. All its positive norms are multiples of four. The simple reflections are adjacent coordinate swaps and a sign change of the fourth coordinate, as verified in the same earlier proof; they preserve \(2\mathbb Z^4\). Hence the orbit of the long simple vector lies in \(M\), and \(x\) fails its parity condition.

Taking positive and negative parts of simple coefficients preserves \(M\). Applying (W.7) to a mixed-sign member of \(M\) gives norm at least eight, excluding every long root of norm four. This is the additional bound that ordinary evenness alone cannot give. □

## X. Changing a simple basis explicitly

**Exercise X.1 (finite maximization in \(A_2\)).** In W.1 with \(m=1\), let \(v=-a\). Find its positive simple basis, a reflection word carrying it to \(\{a,b\}\), and an actual score-maximizing orbit vector for the argument of P.2.

**Solution.** The pairings of \(v\) with \(a,b,a+b\) are \(-2,1,-1\). Thus \(v\) is regular, and its positive roots are \(b,-a,-a-b\). Their indecomposable members are
\[
\Delta_v=\{b,-a-b\},
\]
since their sum is \(-a\). The other two cannot be sums of roots from this three-member positive set, as their coefficients show.

Put \(\eta=a+b\); its pairings with \(a,b\) are both one. By W.1 the orbit of \(v\) is the six roots. Their scores against \(\eta\) are
\[
\begin{array}{c|rrrrrr}
x&a&b&a+b&-a&-b&-a-b\\ \hline
(x,\eta)&1&1&2&-1&-1&-2.
\end{array}
\]
The unique maximum is \(a+b\). The word \(w=s_bs_a\), with its rightmost factor acting first, sends
\[
v=-a\ \longmapsto\ a\ \longmapsto\ a+b.
\]
On the positive basis it sends \(b\mapsto a+b\mapsto a\) and \(-a-b\mapsto-b\mapsto b\). Thus \(w\Delta_v=\{a,b\}\), realizing both the maximum and the basis conjugacy of P.2. □

**Exercise X.2 (a \(B_2\) basis change and a singular vector).** Use \(S_2\) in W.1. In coefficient coordinates let \(\eta=S_2^{-1}(1,1)^t\) and \(v=s_a\eta\). Determine its simple basis and arrow. Then use \(v_0=-a\) to test whether a maximizing orbit vector must have strictly positive simple pairings without regularity.

**Solution.** Solving gives \(\eta=(1,3/2)\), and (W.2) gives \(v=(1/2,3/2)\). Its pairings with \(a,b\) are \((-1,2)\), hence with \(a+b,a+2b\) they are \(1,3\). All are nonzero. Its positive roots are
\[
-a,\quad b,\quad a+b,\quad a+2b,
\]
with indecomposable basis \(a'=-a,\ b'=a+b\). Indeed \(b=a'+b'\) and \(a+2b=a'+2b'\), and a unit coefficient vector cannot be a sum of two nonzero nonnegative coefficient vectors. These basis vectors are \(s_aa,s_ab\), so their Gram matrix is again \(S_2\) and their row Cartan matrix is again \(\begin{pmatrix}2&-1\\-2&2\end{pmatrix}\), of type \(B_2\). Their squared lengths are \(4,2\); the arrow still points to the short vector \(b'\).

For \(v_0=-a\), the pairings with \(a,b\) are \((-4,2)\), and \((v_0,a+2b)=0\). Thus it is singular. Its orbit is the four long roots \(\pm a,\pm(a+2b)\), by the reflection table. Their scores against \(\eta\) are \(1,-1,3,-3\), respectively. The maximum \(a+2b\) has simple pairings \((0,2)\), not two strictly positive values. This exhibits exactly the missing strictness in the proof of P.2 when regularity is dropped. □

**Exercise X.3 (the disconnected rank-two system and its scales).** Explain the usual notation \(D_2\) and the length-scale freedom that its Cartan matrix cannot detect.

**Solution.** In an orthonormal coordinate plane the roots \(\{\pm e_1\pm e_2\}\) are
\[
\{\pm\beta,\pm\gamma\},\qquad
\beta=e_1-e_2,\quad\gamma=e_1+e_2.
\]
The two displayed vectors are perpendicular, each of squared length two. Their reflections change the sign in their own line and fix the other, so this is \(A_1\) orthogonal to \(A_1\), with Cartan matrix \(\operatorname{diag}(2,2)\). Thus it lies outside the connected \(D_n\), \(n\geq4\), range in O.2.

Replace \(\beta,\gamma\) independently by \(s\beta,t\gamma\), with \(s,t>0\). Orthogonality, reflection closure and Cartan matrix stay unchanged; the squared lengths become \(2s^2,2t^2\). Varying \(s/t\) changes their relative length. A single global scale would preserve that ratio, up to interchanging the two lines, so it cannot account for all these possibilities. This is the componentwise scale freedom proved in O.1 and P.1. □

## Y. Matrix algebras, roots and the Killing normalization

Write \(E_{ij}\) for the matrix with a single entry one in position \(i,j\). Matrix multiplication gives
\[
E_{ij}E_{kl}=\delta_{jk}E_{il},\qquad
[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}.
\tag{Y.1}
\]

**Exercise Y.1 (the \(A_2\) algebra and inverse Killing metrics).** Realize the \(A_2\) presentation by traceless \(3\times3\) matrices and prove that the map is an isomorphism. Compute the Killing and inverse root Gram matrices in ranks one and two, including their positive real Cartan spaces.

**Solution.** Put
\[
\begin{gathered}
e_1=E_{12},\quad e_2=E_{23},\quad
f_1=E_{21},\quad f_2=E_{32},\\
h_1=\operatorname{diag}(1,-1,0),\qquad
h_2=\operatorname{diag}(0,1,-1).
\end{gathered}
\tag{Y.2}
\]
For diagonal \(H\), (Y.1) gives \([H,E_{ij}]=(H_{ii}-H_{jj})E_{ij}\). This proves the \(h,e\) and \(h,f\) relations for the type \(A_2\) matrix \(\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\). The \(h_i\) commute; (Y.1) gives \([e_i,f_j]=\delta_{ij}h_i\). Also
\[
[e_1,e_2]=E_{13},\qquad [f_1,f_2]=-E_{31}.
\]
Both \(e_1,e_2\) commute with \(E_{13}\), and both \(f_1,f_2\) commute with \(E_{31}\), again by (Y.1). These are all the Serre relations. Hence the presentation maps to \(\mathfrak{sl}_3(\mathbb C)\).

Its image contains all six off-diagonal matrix units and the two independent diagonal vectors \(h_1,h_2\). These form a basis of the traceless matrices: an arbitrary diagonal trace-zero vector is a unique combination of the latter two, independently of the six off-diagonal entries. Thus the map is surjective onto a space of dimension eight. W.1 gives six \(A_2\) roots, and T.3 gives presentation dimension \(2+6=8\); surjectivity and equality of finite dimensions prove the isomorphism. The same argument in rank one sends \(e,f,h\) to \(E_{12},E_{21},\operatorname{diag}(1,-1)\) and gives the three-dimensional algebra \(\mathfrak{sl}_2\).

Here is a direct trace computation of the Killing form. On all \(n\times n\) matrices let \(L_X(Z)=XZ\), \(R_Y(Z)=ZY\). Using the basis \(E_{ij}\), the diagonal coefficient in \(L_XR_Y(E_{ij})\) is \(X_{ii}Y_{jj}\). Hence
\[
\begin{gathered}
\operatorname{tr}_{\mathrm{End}}(L_XR_Y)
=\operatorname{tr}X\,\operatorname{tr}Y,\\
\operatorname{tr}_{\mathrm{End}}(L_XL_Y)
=n\operatorname{tr}(XY),\qquad
\operatorname{tr}_{\mathrm{End}}(R_XR_Y)
=n\operatorname{tr}(YX).
\end{gathered}
\]
The last two follow by applying the same matrix-unit calculation to left or right multiplication by \(XY\) or \(YX\). Since \(\operatorname{ad}X=L_X-R_X\), expansion gives \(2n\operatorname{tr}(XY)-2\operatorname{tr}X\operatorname{tr}Y\). For traceless \(X,Y\), the decomposition of all matrices as \(\mathfrak{sl}_n\oplus\mathbb CI\) is invariant under both adjoints, and their action on \(\mathbb CI\) is zero. Their trace on the traceless summand is therefore
\[
K(X,Y)=2n\operatorname{tr}(XY).
\tag{Y.3}
\]

In rank one, \(K(h,h)=8\) and the root satisfies \(\alpha(h)=2\). Its Killing-dual vector is \(T_\alpha=h/4\), so
\[
(\alpha,\alpha)=K(T_\alpha,T_\alpha)=\frac12,
\qquad h_\alpha=\frac{2T_\alpha}{(\alpha,\alpha)}=h.
\tag{Y.4}
\]
The real line \(\mathbb Rh\) is positive, while \(K(ih,ih)=-8\). The imaginary line is therefore not the positive real Cartan space of Q.5.

For (Y.2), put \(C=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\). The Cartan Gram matrix and its inverse are
\[
K_{\mathfrak h}=6C=
\begin{pmatrix}12&-6\\-6&12\end{pmatrix},
\qquad
K_{\mathfrak h}^{-1}=\frac1{18}
\begin{pmatrix}2&1\\1&2\end{pmatrix}.
\tag{Y.5}
\]
The simple-root covectors have coordinate rows \((2,-1)\) and \((-1,2)\), so their Gram matrix in the inverse Killing metric is
\[
C K_{\mathfrak h}^{-1}C^t
=\frac16 C
=\begin{pmatrix}1/3&-1/6\\-1/6&1/3\end{pmatrix}.
\tag{Y.6}
\]
This is one sixth of the common abstract \(A_2\) normalization \(C\). In particular \(T_{\alpha_i}=h_i/6\), and the normalized coroot remains \(h_i\). On \(\mathbb Rh_1+\mathbb Rh_2\), the form is \(6\) times the sum of the squares of the real diagonal entries, positive except at zero. It is a real form of the complex Cartan plane. This distinguishes its intrinsic Killing scale from freely chosen abstract root-length units. □

**Exercise Y.2 (strings, parity and proportional roots).** Compute the actual \(\alpha\)-string through \(\beta\) for \(\alpha=\varepsilon_1-\varepsilon_2\), \(\beta=\varepsilon_2-\varepsilon_3\) in (Y.2). Check the endpoint sign for a short root of squared length two and a long root of squared length six with inner product \(-3\). Explain the even-weight zero-multiplicity argument and the additional step excluding nonintegral proportional roots.

**Solution.** The root vectors for \(\beta,\beta+\alpha\) are \(E_{23},E_{13}\). Put \(E=\operatorname{ad}E_{12}\), \(F=\operatorname{ad}E_{21}\), \(H=\operatorname{ad}h_1\). Formula (Y.1) gives
\[
\begin{array}{c|cc}
&E_{23}&E_{13}\\ \hline
H&-E_{23}&E_{13}\\
E&E_{13}&0\\
F&0&E_{23}.
\end{array}
\tag{Y.7}
\]
The roots of the matrix Cartan are exactly the six \(\varepsilon_i-\varepsilon_j\), because the six off-diagonal units have those weights and the diagonal subspace has weight zero. Thus the string is exactly \(\beta,\beta+\alpha\), with lower and upper parameters \(r=0,q=1\). The formula \(r-q=\beta(h_\alpha)\) gives \(-1\), and \(s_\alpha\beta=\beta+\alpha\).

For the other calculation, take \(\alpha=b,\beta=a\) in the \(G_2\) system W.1. The entire string is \(\beta,\beta+\alpha,\beta+2\alpha,\beta+3\alpha\). Its coroot pairing is \(2(-3)/2=-3\), so \(r-q=0-3=-3\) and \(s_\alpha\beta=\beta+3\alpha\). These scalar checks concern the verified abstract root system. The existence of a Lie algebra carrying it is supplied separately by T.1–U.1; a list of four root vectors in a Euclidean plane does not by itself define brackets.

By Q.1, an irreducible finite-dimensional \(\mathfrak{sl}_2\) module \(V_m\) has weights \(m,m-2,\ldots,-m\), each once. A completely reducible module having only even weights can contain only even \(m\). Every such summand, including \(V_0\), contributes one zero weight. If the zero space has dimension one and a \(V_2\) summand is present, that is the entire module. Merely knowing that the zero space has dimension one, without the given \(V_2\), would also permit a single \(V_4\) or another even highest weight.

Finally, suppose \(\delta=c\alpha\) is a proportional root and root-space theory has excluded integral multiples of magnitude at least two for every root. The actual coroot triples and weight integrality in Q.3 give
\[
2c\in\mathbb Z,\qquad 2/c\in\mathbb Z.
\]
The first relation already makes \(c\) real and rational. The two nonzero integers have product four, so
\(c\in\{\pm1/2,\pm1,\pm2\}\).
The values \(\pm2\) are excluded by the hypothesis for \(\alpha\); the values \(\pm1/2\) are excluded by the same hypothesis for \(\delta\), since then \(\alpha=\pm2\delta\). Thus only opposite or equal roots remain. Excluding integral multiples in one chosen root direction without this reciprocal argument would not establish reducedness. □

**Exercise Y.3 (the solvable and zero cases).** For \([H,X]=X\) in a two-dimensional algebra, compute Cartan weights and the Killing form and explain the semisimplicity hypothesis. State the output for the zero algebra.

**Solution.** In the ordered basis \(H,X\), the adjoint matrices are
\[
\operatorname{ad}H=\begin{pmatrix}0&0\\0&1\end{pmatrix},
\qquad
\operatorname{ad}X=\begin{pmatrix}0&0\\-1&0\end{pmatrix}.
\]
They give Killing matrix \(\begin{pmatrix}1&0\\0&0\end{pmatrix}\). Relative to \(\mathbb CH\), the weights are zero on \(H\) and \(\alpha(H)=1\) on \(X\); the negative weight is absent. The derived algebra is \(\mathbb CX\), whose own bracket is zero, so the whole algebra is solvable and its Killing form degenerate. Its one nonzero weight cannot be a root system: reflection in it would require its negative. This verifies concretely why Q.2–Q.5 require semisimplicity.

For the zero algebra, the Cartan, its positive real form and its dual are all zero, and the root set and diagram are empty. Root production alone gives these data; the converse construction and uniqueness for all diagrams require S.1–U.1. The empty presentation there indeed has no generators and is the zero algebra by S.1. Neither conclusion is a classification of real forms or Hermitian pairs. □

## Z. Calculations inside the presentation

**Exercise Z.1 (vacuum values, an overlap and the Serre exponent).** In (S.7), calculate \(E_iF_i^k1\) and explain the use of variable vacuum values. Resolve both reductions of \(ehf\) in the ordered-word theorem for \(f<h<e\). For the long-first \(G_2\) Cartan matrix, identify the two positive Serre powers and the vanishing coefficient in (T.4).

**Solution.** For \(k\geq1\), commute \(E_i\) past the \(k\) copies of \(F_i\), using \([E_i,F_i]=H_i\) and \(E_i1=0\). The \(H_i\) at a suffix of length \(j\) acts there by \(t_i-2j\), by (S.7). Thus
\[
E_iF_i^k1
=\sum_{j=0}^{k-1}(t_i-2j)F_i^{k-1}1
=k(t_i-k+1)F_i^{k-1}1.
\tag{Z.1}
\]
At \(k=0\) the result is zero. If \(\sum c_ih_i\) were zero, its action on the polynomial vacuum would give the zero polynomial \(\sum c_it_i\), forcing every coefficient to vanish. One fixed evaluation \(t_i=\lambda_i\) would give only the scalar equation \(\sum c_i\lambda_i=0\); in general that cannot distinguish all coefficient vectors. The variable vacuum, or all evaluations together, proves the required independence.

The relations for sorting the three rank-one generators are
\[
eh=he-2e,\qquad hf=fh-2f,\qquad ef=fe+h.
\]
Sorting the left inverted pair first gives
\[
\begin{aligned}
ehf&=(he-2e)f\\
&=h(fe+h)-2(fe+h)\\
&=(fh-2f)e+h^2-2fe-2h\\
&=fhe+h^2-4fe-2h.
\end{aligned}
\]
Sorting the right inverted pair first gives
\[
\begin{aligned}
ehf&=e(fh-2f)\\
&=(fe+h)h-2(fe+h)\\
&=f(he-2e)+h^2-2fe-2h\\
&=fhe+h^2-4fe-2h.
\end{aligned}
\tag{Z.2}
\]
All terms in the final expression are ordered. This is the explicit rank-one instance of the general Jacobi overlap in S.1.

For \(A=\begin{pmatrix}2&-1\\-3&2\end{pmatrix}\), the positive relations are
\[
(\operatorname{ad}e_1)^2e_2=0,\qquad
(\operatorname{ad}e_2)^4e_1=0.
\tag{Z.3}
\]
The short-root row is the second, so it carries the fourth power. The coefficient \(-m(a_{ij}+m-1)\) in (T.4) is respectively \(-2(-1+2-1)=0\) and \(-4(-3+4-1)=0\). Interchanging these exponents would not give the prescribed row convention. The negative relations and their coefficients follow from the signed involution (S.8). □

**Exercise Z.2 (degrees and the sign of a reflected vector).** Explain the different reasons that type \(A_2\) has no degree \(a-b\) or \(2a\) in the presentation. For \(\gamma=a+b\), compute its reflection in \(a\) and the corresponding action on \(e_{12}=[e_1,e_2]\) in (Y.2).

**Solution.** The degree \(a-b\) has both signs, so the exact triangular decomposition T.1 excludes it. The degree \(2a\) is in the positive cone, but its positive bracketed words use only \(e_1\). Their length is at least two, so they vanish by \([e_1,e_1]=0\) and induction on bracketed words, as proved in T.1. The two exclusions use different parts of that proof.

The root \(\gamma=a+b\) has \(\gamma(h_1)=2-1=1\), hence \(s_a\gamma=b\); its height drops from two to one. In (Y.2), \(e_{12}=E_{13}\). Put \(E=\operatorname{ad}E_{12}\), \(F=\operatorname{ad}E_{21}\). Formula (Y.1) gives \(EE_{13}=0\), \(FE_{13}=E_{23}\), \(FE_{23}=0\), and \(EE_{23}=E_{13}\). Applying (T.5) from right to left therefore gives
\[
E_{13}
\ \xmapsto{\exp E}\ E_{13}
\ \xmapsto{\exp(-F)}\ E_{13}-E_{23}
\ \xmapsto{\exp E}\ -E_{23}.
\tag{Z.4}
\]
Thus the reflected space is the \(b\)-space, but this particular normalized vector acquires a minus sign. Root reflection and the scalar choice of a root vector are different data. □

**Exercise Z.3 (disconnected and exceptional existence, rescaling and the boundary).** Reconstruct \(\operatorname{diag}(2,2)\), identify the nonvanishing argument for the \(E_8\) presentation, and show that rescaling simple root vectors does not change the presented algebra. Include the empty case and the failure of the finite-type proof for (W.5).

**Solution.** Each rank-one presentation has basis \(e,f,h\) and is \(\mathfrak{sl}_2\) by Y.1. For \(\operatorname{diag}(2,2)\), every cross-component generator bracket is zero: the positive and negative Serre exponents are one, while all other cross brackets vanish in (S.5). The two commuting inclusions and projections in T.4 therefore identify the presentation exactly with
\[
\mathfrak{sl}_2\oplus\mathfrak{sl}_2,
\]
of dimension six. Its four roots are the two opposite pairs in X.3. This differs from the connected type \(A_2\) algebra of dimension eight and six roots in Y.1.

For \(E_8\), the positive matrix is provided by O.2, and a root system with that precise diagram is proved in [De Rham decomposition AE.4](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-4) and transferred to the row convention in P.1. Its 240 roots comprise \(4\binom82=112\) two-coordinate vectors and \(2^7=128\) half-coordinate vectors with the proved parity condition. The actual Lie-algebra existence argument is the following chain of proved constructions. S.1 constructs the formal free Lie algebra and proves its tensor embedding. The module of S.2 keeps all Cartan generators independent. T.1 identifies the exact Serre ideal and shows that every simple generator survives. T.2 realizes root reflections by automorphisms. T.3 then forces exactly the finite root set, with one-dimensional root spaces, and T.4 proves simplicity. Consequently the \(E_8\) algebra exists and has dimension \(8+240=248\). Counting an abstract orbit alone would prove none of the presentation's noncollapse or Jacobi-compatible bracket assertions.

For arbitrary nonzero \(c_i\in\mathbb C\), the substitutions
\[
e_i\longmapsto c_i e_i,\qquad
f_i\longmapsto c_i^{-1}f_i,\qquad
h_i\longmapsto h_i
\tag{Z.5}
\]
preserve (S.5). In particular the factors cancel in \([e_i,f_i]=h_i\); off-diagonal mixed brackets remain zero. A positive Serre expression of exponent \(m\) is multiplied by \(c_i^m c_j\), and a negative one by \(c_i^{-m}c_j^{-1}\), so both remain zero. The universal property gives an endomorphism, and substitution of the reciprocal scalars gives its inverse. It is therefore an automorphism. This accounts for arbitrary choices of nonzero simple root vectors in the one-dimensional spaces; normalization of the opposite vector to keep \(h_i\) fixes its reciprocal factor.

For rank zero the free presentation is zero, with empty root system and no components. For (W.5), the determinant and positive definiteness required in T.1 and T.3 fail; W.2 exhibits the null vector and an infinite formal reflection orbit explicitly. The identification of degrees with distinct weights and the positive-norm height argument cannot be used there. The proof consequently makes no finite-type or semisimplicity assertion for that boundary matrix. □

## AA. Compact real forms and real semisimple algebras

A **conjugation** of a complex Lie algebra \(\mathfrak g\) is an antilinear bracket-preserving map \(\rho\) with \(\rho^2=1\). Its fixed algebra \(\mathfrak g^\rho\) is a real form: every \(X\in\mathfrak g\) has the unique expression
\[
X=\frac{X+\rho X}{2}
+i\,\frac{X-\rho X}{2i}
\tag{AA.1}
\]
with both displayed summands before multiplication by \(i\) fixed by \(\rho\). Conversely a real form determines a conjugation by \(U+iV\mapsto U-iV\). We call a real form **compact** if its real Killing form is negative definite. This terminology agrees with compact groups by F.2. In this part \(\mathfrak g\) is a finite-dimensional complex semisimple algebra, and \(K\) is its complex Killing form.

**Theorem AA.1 (construction of a compact real form).** In the Serre generators of U.1, the assignments
\[
\tau h_i=-h_i,\qquad \tau e_i=-f_i,\qquad \tau f_i=-e_i
\tag{AA.2}
\]
extend antilinearly to a conjugation. Its fixed algebra \(\mathfrak u\) is a compact real form. The form
\[
B(X,Y)=-K(X,\tau Y)
\tag{AA.3}
\]
is a positive-definite Hermitian form, linear in its first argument. In particular, every complex semisimple algebra has a compact real form.

**Proof.** U.1 identifies \(\mathfrak g\) with the algebra defined by the real-coefficient Serre relations. Conjugating coefficients of formal words therefore gives an antilinear involution \(\kappa\) fixing every generator. The interchange \(\omega\) of (S.8) preserves the positive and negative Serre relations as well as the preliminary relations: each positive iterated bracket goes to its negative counterpart with the same exponent. Thus it descends to this quotient. Compose \(\kappa\), \(\omega\), and the scaling \(e_i\mapsto-e_i,\ f_i\mapsto-f_i,\ h_i\mapsto h_i\) from Z.3. This gives (AA.2), preserves the bracket, and has square one on the generators, hence on the whole algebra. Formula (AA.1) proves the real-form assertion.

For any conjugation \(\eta\), use a real basis of its fixed algebra as a complex basis of \(\mathfrak g\). The matrix of \(\operatorname{ad}(\eta X)\) is the entrywise conjugate of that of \(\operatorname{ad}X\). Taking traces gives
\[
K(\eta X,\eta Y)=\overline{K(X,Y)}.
\tag{AA.4}
\]
Symmetry of \(K\) and (AA.4) show that \(B\) is Hermitian. Killing invariance gives, without yet assuming positivity,
\[
B([Z,X],Y)=B(X,[-\tau Z,Y]).
\tag{AA.5}
\]
Indeed \(K([Z,X],\tau Y)=K(X,[\tau Y,Z])\), and applying \(\tau\) to \([-\tau Z,Y]\) gives \([\tau Y,Z]\).

Let \(\mathfrak h\) be the Cartan spanned by the \(h_i\), and let \(\mathfrak h_{\mathbb R}\) be their real span. By Q.5, \(K\) is positive definite on this real span. Since \(\tau=-1\) there, (AA.3) is its positive Hermitian extension to \(\mathfrak h\). The operators \(\operatorname{ad}h_i\) are self-adjoint for \(B\) by (AA.5), and all their root eigenvalues are real by Q.5. The simultaneous weight spaces with different weights are \(B\)-orthogonal: if \(\lambda\ne\mu\) are their eigenvalues for some \(h_i\), then
\((\lambda-\mu)B(X,Y)=0\). The roots are determined by their values on the \(h_i\), so this applies to distinct root spaces and to a root space paired with \(\mathfrak h\).

On a simple root line,
\[
B(e_i,e_i)=K(e_i,f_i)=\tfrac12K(h_i,h_i)>0.
\tag{AA.6}
\]
The equality follows from
\(K([e_i,f_i],h_i)=K(e_i,[f_i,h_i])=2K(e_i,f_i)\).
We now prove positivity on every positive root line by height induction. For a nonsimple positive root \(\beta\), the height argument used in U.1 supplies a simple root \(\alpha_i\) such that \(\gamma=\beta-\alpha_i\) is a positive root. These two roots are not proportional to \(\alpha_i\), since the system is reduced. By Q.4, the \(\alpha_i\)-string through \(\gamma\) is an irreducible rank-one module. Put \(E=\operatorname{ad}e_i\), \(F=\operatorname{ad}f_i\). For a nonzero \(v\in\mathfrak g_\gamma\), the root \(\beta\) exists, so \(Ev\ne0\). In the convention (Q.3), \(v\) is a multiple of some \(v_j\) with \(1\leq j\leq m\), and
\[
FEv=j(m-j+1)v.
\]
By (AA.5), \(E\) and \(F\) are adjoint to one another. Consequently
\[
B(Ev,Ev)=B(v,FEv)
=j(m-j+1)B(v,v)>0.
\tag{AA.7}
\]
Here the scalar is real and positive, and the induction hypothesis applies to the smaller height of \(\gamma\). Root spaces have dimension one by Q.3, so this proves their positivity.

For \(T\in\mathfrak h_{\mathbb R}\) and \(X\in\mathfrak g_\alpha\), applying \(\tau\) to \([T,X]=\alpha(T)X\) shows that \(\tau X\in\mathfrak g_{-\alpha}\). Also
\[
B(\tau X,\tau X)=-K(\tau X,X)=B(X,X).
\]
Thus the negative root lines are positive as well. Their orthogonal direct sum with \(\mathfrak h\) is all of \(\mathfrak g\), proving positive definiteness of \(B\).

On \(\mathfrak u=\mathfrak g^\tau\), \(B=-K\). The real Killing form of \(\mathfrak u\) is exactly \(K|_{\mathfrak u}\), because the real adjoint matrices in a real basis also compute the complex trace after complexification. It is therefore negative definite. [Symmetric spaces H.5–H.6](symmetric-spaces.md#theorem-h-5) give semisimplicity of \(\mathfrak u\). The construction in C.4 realizes it as the Lie algebra of \(\operatorname{Aut}(\mathfrak u)^0\): the bracket equations define a closed matrix subgroup, and its derivation algebra is \(\operatorname{ad}\mathfrak u\simeq\mathfrak u\). Now F.2 makes this group and its simply connected cover compact. For \(\mathfrak g=0\), all assertions hold on the zero vector space and the group is a point. □

**Lemma AA.2 (powers of a positive automorphism).** Suppose \(A\) is a complex Lie-algebra automorphism which is positive and self-adjoint for a positive-definite Hermitian form. For every real \(t\), the spectral power \(A^t\) is an automorphism, and \(D=\log A\) is a derivation. If the algebra is semisimple, then
\[
D=\operatorname{ad}Z,\qquad A^t=\exp(t\operatorname{ad}Z)
\tag{AA.8}
\]
for a unique \(Z\). If an antilinear involution \(\eta\) satisfies \(\eta A\eta=A^{-1}\), then \(\eta A^t\eta=A^{-t}\). If it commutes with \(A\), it commutes with \(A^t\) and \(D\).

**Proof.** K.1 supplies an orthogonal direct sum of eigenspaces \(V_\lambda\) with \(\lambda>0\). Bracket preservation gives
\[
[V_\lambda,V_\mu]\subseteq V_{\lambda\mu},
\tag{AA.9}
\]
where the bracket is zero if \(\lambda\mu\) is absent from the spectrum. The positive real exponential \(E:\mathbb R\to\mathbb R_{>0}\), its smooth inverse \(\log\), and \(E(s+t)=E(s)E(t)\) are constructed in [Geodesics A.4](geodesics-normal-coordinates-and-curvature.md#example-a-4). Define \(\lambda^t=E(t\log\lambda)\) for \(\lambda>0\). Applying the inverse logarithm to \(E(\log\lambda+\log\mu)=\lambda\mu\) proves \(\log(\lambda\mu)=\log\lambda+\log\mu\); applying the addition identity again gives \((\lambda\mu)^t=\lambda^t\mu^t\). Define \(A^t\) and \(D\) on \(V_\lambda\) by these real scalars \(\lambda^t\) and \(\log\lambda\). Since \((\lambda\mu)^t=\lambda^t\mu^t\), (AA.9) proves that \(A^t\) preserves the bracket; \(A^{-t}\) is its inverse. Since \(\log(\lambda\mu)=\log\lambda+\log\mu\), the same calculation proves the derivation rule for \(D\). [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6) makes this derivation uniquely inner, and the exponential on each eigenspace proves (AA.8).

If \(\eta A\eta=A^{-1}\), then \(\eta\) carries \(V_\lambda\) to \(V_{\lambda^{-1}}\). The eigenvalues are real, so its antilinearity does not change them; the power identity follows on each eigenspace. If \(\eta\) commutes with \(A\), it preserves every \(V_\lambda\), and commutes with every indicated real scalar operator there. This also covers the zero algebra. □

**Theorem AA.3 (conjugacy of compact real forms).** Any two compact conjugations \(\tau_0,\tau_1\) of \(\mathfrak g\) are conjugate by an automorphism of the form \(\exp(\operatorname{ad}Z)\). If both commute with another conjugation \(\rho\), the conjugating automorphism can be chosen to preserve \(\mathfrak g^\rho\), with \(Z\in\mathfrak g^\rho\).

**Proof.** For a compact conjugation \(\tau_i\), put \(B_i(X,Y)=-K(X,\tau_iY)\). If \(X=U+iV\) with \(U,V\in\mathfrak g^{\tau_i}\), symmetry and reality of \(K\) give
\[
B_i(X,X)=-K(U,U)-K(V,V).
\tag{AA.10}
\]
Thus \(B_i\) is positive definite, as well as Hermitian by (AA.4). Let \(A=\tau_0\tau_1\), a complex automorphism. Then
\[
B_1(X,Y)=B_0(X,AY).
\tag{AA.11}
\]
Hermitian symmetry on both sides makes \(A\) self-adjoint for \(B_0\), and (AA.10) makes it positive. Also
\[
\tau_0 A\tau_0=A^{-1},\qquad
\tau_1=A^{-1}\tau_0.
\tag{AA.12}
\]
Apply AA.2. The automorphism \(\phi=A^{-1/2}\) satisfies
\[
\phi\tau_0\phi^{-1}
=A^{-1/2}\tau_0A^{1/2}
=A^{-1}\tau_0=\tau_1.
\tag{AA.13}
\]
It is the exponential of the inner derivation \(-\tfrac12\log A\), which proves the first assertion.

If \(\rho\) commutes with both conjugations, it commutes with \(A\) and, by AA.2, with \(\log A\) and \(\phi\). Write \(\log A=\operatorname{ad}W\). Conjugating this equality by \(\rho\) gives \(\operatorname{ad}(\rho W)=\operatorname{ad}W\). The zero centre in [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6) gives \(\rho W=W\). Hence \(Z=-W/2\) lies in the real fixed algebra, and \(\phi\) preserves that algebra. □

**Theorem AA.4 (a compact conjugation commuting with any real form).** For every conjugation \(\rho\) of \(\mathfrak g\), there is a compact conjugation \(\tau\) commuting with \(\rho\). Writing \(\mathfrak u=\mathfrak g^\tau\) and \(\sigma=(\rho\tau)|_{\mathfrak u}\), one has an involution of the compact real algebra and a direct sum
\[
\mathfrak g^\rho=\mathfrak u_+\oplus i\mathfrak u_-,
\qquad \mathfrak u_\pm=\ker(\sigma\mp1).
\tag{AA.14}
\]
The real Killing form is negative on \(\mathfrak u_+\), positive on \(i\mathfrak u_-\), and makes them orthogonal.

**Proof.** Choose \(\tau_0\) by AA.1 and put \(\tau_1=\rho\tau_0\rho\). Its fixed real algebra is \(\rho(\mathfrak g^{\tau_0})\). Formula (AA.4) shows that \(K\) is again real and negative definite there, so \(\tau_1\) is compact. As in AA.3, \(A=\tau_0\tau_1\) is positive self-adjoint for \(B_0\). In addition to (AA.12), it satisfies
\[
\rho A\rho=\tau_1\tau_0=A^{-1}.
\tag{AA.15}
\]
Define
\[
\tau=A^{-1/2}\tau_0
=A^{-1/4}\tau_0A^{1/4}.
\tag{AA.16}
\]
The second identity follows from AA.2. It exhibits \(\tau\) as an automorphic conjugate of \(\tau_0\), hence as a compact conjugation. Using (AA.15) and \(\rho\tau_0\rho=\tau_1=A^{-1}\tau_0\), one obtains
\[
\rho\tau\rho=A^{1/2}\tau_1
=A^{-1/2}\tau_0=\tau.
\tag{AA.17}
\]
This proves commutation.

The complex-linear map \(\rho\tau\) now has square one and preserves \(\mathfrak u\); its restriction \(\sigma\) is a real Lie-algebra involution. For \(U,V\in\mathfrak u\), the equation \(\rho(U+iV)=U+iV\) is precisely \(\sigma U=U,\ \sigma V=-V\). This proves (AA.14). An automorphism preserves \(K\), because it conjugates each adjoint operator. Consequently the two eigenspaces of \(\sigma\) are \(K\)-orthogonal. Negativity on \(\mathfrak u\) and \(K(iV,iV)=-K(V,V)\) prove the signs.

On the real algebra \(\mathfrak g^\rho\), the restriction \(\theta=\tau|_{\mathfrak g^\rho}\) is an involution with these positive and negative eigenspaces. The form \(-K(X,\theta Y)\) is positive definite: for \(X=U+iV\) as above its squared value is \(-K(U,U)-K(V,V)\). Thus \(\theta\) is a **Cartan involution**, meaning an involution with this positivity property. All bracket inclusions follow directly from multiplying its eigenvalue signs. Semisimplicity of the real algebra follows from [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6), since its complexification is \(\mathfrak g\). □

**Theorem AA.5 (real forms are classified by compact involutions).** Fix a compact real form \(\mathfrak u\) of \(\mathfrak g\). Sending an involution \(\sigma\in\operatorname{Aut}_{\mathbb R}(\mathfrak u)\) to
\[
\mathfrak r_\sigma=\mathfrak u_+\oplus i\mathfrak u_-
\tag{AA.18}
\]
induces a bijection between conjugacy classes of involutions under \(\operatorname{Aut}_{\mathbb R}(\mathfrak u)\) and isomorphism classes of real forms of \(\mathfrak g\). The Killing signature of \(\mathfrak r_\sigma\), in the order (positive, negative), is
\[
(\dim_{\mathbb R}\mathfrak u_-,\,\dim_{\mathbb R}\mathfrak u_+).
\tag{AA.19}
\]

**Proof.** Extend \(\sigma\) complex linearly to \(\mathfrak g\), and let \(\tau\) be its fixed compact conjugation. The extension commutes with \(\tau\), since both preserve the decomposition \(\mathfrak u+i\mathfrak u\). Thus \(\rho_\sigma=\sigma_{\mathbb C}\tau\) is a conjugation, whose fixed algebra is (AA.18). Its bracket is closed: even-even and odd-odd brackets are even, and even-odd brackets are odd; the two factors of \(i\) in an odd-odd bracket only change its sign. The orthogonality and signs in (AA.19) follow exactly as in AA.4. Compactly conjugate involutions clearly give isomorphic real forms by the complex extension of the conjugating map.

Conversely, AA.4 supplies for any real conjugation \(\rho\) a commuting compact conjugation \(\tau'\). By AA.3, choose a complex automorphism carrying \(\tau'\) to \(\tau\). It carries \(\rho\) to a conjugation commuting with \(\tau\). Their product then restricts to an involution of \(\mathfrak u\), and its real form is the image of \(\mathfrak g^\rho\). This proves surjectivity.

For injectivity, suppose \(f:\mathfrak r_{\sigma_1}\to\mathfrak r_{\sigma_2}\) is a real Lie-algebra isomorphism. By (AA.1), its complex-linear extension \(\Phi\) is an automorphism of \(\mathfrak g\) satisfying
\(\Phi\rho_{\sigma_1}\Phi^{-1}=\rho_{\sigma_2}\).
The compact conjugation \(\tau'=\Phi\tau\Phi^{-1}\) commutes with \(\rho_{\sigma_2}\), as does \(\tau\). The relative assertion of AA.3 therefore supplies an automorphism \(\psi\), commuting with \(\rho_{\sigma_2}\), such that \(\psi\tau'\psi^{-1}=\tau\). Now \(F=\psi\Phi\) preserves \(\tau\), hence preserves \(\mathfrak u\), and it still intertwines the two \(\rho_{\sigma_i}\). Multiplying by \(\tau\) shows that \(F|_{\mathfrak u}\) conjugates \(\sigma_1\) to \(\sigma_2\). This proves injectivity, including semisimple algebras with several simple factors and the zero algebra. □

## AB. Involutions and invariant simple roots

Let \(\mathfrak u\) be a compact real semisimple algebra, let \(\mathfrak g=\mathfrak u\otimes_{\mathbb R}\mathbb C\), and let \(\tau\) be its conjugation. Write \(K\) for the complex Killing form. For an involution \(\sigma\) of \(\mathfrak u\), use the same letter for its complex extension and put
\[
\mathfrak k=\mathfrak u_+,\qquad
\mathfrak p=\mathfrak u_-,\qquad Q=-K|_{\mathfrak u}.
\tag{AB.1}
\]

**Theorem AB.1 (a Cartan adapted to the fixed algebra).** The fixed algebra is reductive:
\[
\mathfrak k=Z(\mathfrak k)\oplus[\mathfrak k,\mathfrak k],
\tag{AB.2}
\]
where the second summand is compact semisimple. There is a \(\sigma\)-stable maximal abelian subalgebra
\(\mathfrak t=\mathfrak t_+\oplus\mathfrak t_-\) of \(\mathfrak u\), with \(\mathfrak t_\pm\subseteq\mathfrak u_\pm\), such that \(\mathfrak t_+\) is maximal abelian in \(\mathfrak k\). Its complexification \(\mathfrak h=\mathfrak t_{\mathbb C}\) is a Cartan subalgebra of \(\mathfrak g\). No root of \((\mathfrak g,\mathfrak h)\) vanishes identically on \(\mathfrak t_+\). One can consequently choose a positive system, and hence a simple-root basis, preserved by \(\sigma\).

**Proof.** Invariance of the positive form \(Q\) gives
\[
[\mathfrak k,\mathfrak k]^{\perp_Q}=Z(\mathfrak k).
\tag{AB.3}
\]
Indeed \(Q(X,[Y,Z])=Q([X,Y],Z)\); its vanishing for all \(Y,Z\in\mathfrak k\) is equivalent to \([X,Y]=0\) for every \(Y\), by positive definiteness on \(\mathfrak k\). Taking orthogonal complements proves (AB.2). Write \(\mathfrak s=[\mathfrak k,\mathfrak k]\). Its centre is zero: an element centralizing \(\mathfrak s\) within \(\mathfrak s\) also centralizes \(Z(\mathfrak k)\), hence belongs to \(Z(\mathfrak k)\cap\mathfrak s=0\). On \(\mathfrak s\), each \(\operatorname{ad}X\) is skew-adjoint for \(Q\). In an orthonormal real basis,
\[
K_{\mathfrak s}(X,X)
=\operatorname{tr}_{\mathfrak s}(\operatorname{ad}X)^2
=-\sum_j\lVert[X,v_j]\rVert_Q^2.
\tag{AB.4}
\]
This is strictly negative for \(X\ne0\), since the centre of \(\mathfrak s\) is zero. [Symmetric spaces H.5](symmetric-spaces.md#theorem-h-5) proves semisimplicity. This proves the first assertion without assuming a theorem about reductive fixed subalgebras.

Choose an abelian subalgebra \(\mathfrak t_+\) of maximal dimension in \(\mathfrak k\). Then \(C_{\mathfrak k}(\mathfrak t_+)=\mathfrak t_+\), since any commuting element could otherwise be adjoined. Choose a subspace \(\mathfrak t_-\subseteq\mathfrak p\) of maximal dimension whose elements commute with one another and with \(\mathfrak t_+\). The direct sum \(\mathfrak t=\mathfrak t_+\oplus\mathfrak t_-\) is abelian and \(\sigma\)-stable. Its centralizer in \(\mathfrak u\) is \(\sigma\)-stable and therefore splits into its two eigenspaces. Its even part lies in \(C_{\mathfrak k}(\mathfrak t_+)=\mathfrak t_+\); an odd element commuting with \(\mathfrak t\) can be adjoined to \(\mathfrak t_-\), so maximality places it in \(\mathfrak t_-\). Thus \(C_{\mathfrak u}(\mathfrak t)=\mathfrak t\).

For \(T\in\mathfrak t\), \(\operatorname{ad}T\) is skew-adjoint for \(Q\); on the complexification it is skew-adjoint for the positive Hermitian form \(B=-K(\,\cdot\,,\tau\,\cdot\,)\) from (AA.10). Hence \(i\operatorname{ad}T\) is self-adjoint. K.1 diagonalizes it. A finite basis of the commuting operators can be simultaneously diagonalized by successively restricting to their invariant eigenspaces. It follows that \(\mathfrak h=\mathfrak t_{\mathbb C}\) is toral, and complexifying the centralizer equation gives \(C_{\mathfrak g}(\mathfrak h)=\mathfrak h\). This is the Cartan definition of Q.2. Its roots take purely imaginary values on \(\mathfrak t\).

The action on roots is \((\sigma\alpha)(H)=\alpha(\sigma^{-1}H)\). If a root \(\alpha\) vanished on \(\mathfrak t_+\), this equation on \(\mathfrak t_+\oplus\mathfrak t_-\) would give \(\sigma\alpha=-\alpha\). For \(0\ne e\in\mathfrak g_\alpha\), the vector
\(e+\sigma e\) is nonzero because its two terms lie in distinct root spaces. It is fixed by \(\sigma\) and commutes with \(\mathfrak t_+\). But
\[
C_{\mathfrak k_{\mathbb C}}(\mathfrak t_+)
=(\mathfrak t_+)_{\mathbb C},
\]
by taking real and imaginary parts in the earlier centralizer equation. This is a subspace of \(\mathfrak h\), whereas \(e+\sigma e\) has only nonzero root components, a contradiction.

There are only finitely many roots. Since their restrictions are nonzero, choose \(T\in\mathfrak t_+\) on none of their kernels; the finite-hyperplane argument is proved in [De Rham decomposition AD.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ad-5). Then the real numbers \(-i\alpha(T)\) are all nonzero. Declare a root positive when this number is positive. Q.5 identifies the real span of the roots with a Euclidean root space, and P.1 proves that such a regular real linear functional defines a positive system with a simple basis. Since \(\sigma T=T\), this functional has the same value on \(\alpha\) and \(\sigma\alpha\). Thus \(\sigma\) preserves positive roots and their indecomposable members, the simple roots. The zero algebra has empty roots and requires no choice of \(T\). □

**Theorem AB.2 (signed diagram representatives).** Every involution of a compact semisimple real algebra admits Serre generators in its complexification for which
\[
\begin{aligned}
\tau h_i&=-h_i,&\tau e_i&=-f_i,&\tau f_i&=-e_i,\\
\sigma h_i&=h_{\pi(i)},&
\sigma e_i&=c_i e_{\pi(i)},&
\sigma f_i&=c_i^{-1}f_{\pi(i)}.
\end{aligned}
\tag{AB.5}
\]
Here \(\pi^2=1\) is a permutation preserving the row Cartan matrix, \(c_i=c_{\pi(i)}=1\) on every two-element orbit, and \(c_i\in\{1,-1\}\) on fixed vertices. Conversely each such diagram permutation and choice of signs defines an involution of the compact form constructed in AA.1. Every compact-involution conjugacy class has at least one of these representatives; distinct choices are not asserted to give distinct classes.

**Proof.** Use the Cartan and simple roots of AB.1. Let \(T_\alpha\in\mathfrak h\) be the \(K\)-dual root vector, defined by \(K(T_\alpha,H)=\alpha(H)\). Since roots are purely imaginary on \(\mathfrak t\), writing \(T_\alpha=U+iV\) with \(U,V\in\mathfrak t\) gives \(K(U,T)=0\) for every \(T\in\mathfrak t\). Negativity on \(\mathfrak t\) gives \(U=0\), so \(T_\alpha\in i\mathfrak t\), where \(K\) is positive definite. In particular
\[
q_\alpha=K(T_\alpha,T_\alpha)>0,\qquad
h_\alpha=2T_\alpha/q_\alpha.
\tag{AB.6}
\]
These are the same coroots as in Q.3–Q.5.

The conjugation \(\tau\) carries \(\mathfrak g_\alpha\) to \(\mathfrak g_{-\alpha}\): for \(T\in\mathfrak t\), conjugating the purely imaginary eigenvalue reverses its sign. Choose \(0\ne e\in\mathfrak g_{\alpha_i}\) and put \(f=-\tau e\). The opposite-root calculation of Q.3 gives
\[
[e,f]=K(e,f)T_{\alpha_i}=B(e,e)T_{\alpha_i}.
\]
Multiplying \(e\) by a positive real scalar \(a\) and \(f\) by the same scalar changes the coefficient to \(a^2B(e,e)\). Choose \(a^2=2/(q_{\alpha_i}B(e,e))\). Then \([e_i,f_i]=h_i=h_{\alpha_i}\) and \(\tau e_i=-f_i\). The generators so obtained satisfy the Serre relations and generate \(\mathfrak g\), by the proof of U.1.

By AB.1, \(\sigma\) permutes the simple roots by an involutive permutation \(\pi\). Invariance of \(K\) gives \(\sigma T_{\alpha_i}=T_{\alpha_{\pi(i)}}\), hence \(\sigma h_i=h_{\pi(i)}\). It also proves
\[
\alpha_j(h_i)=\alpha_{\pi(j)}(h_{\pi(i)}),
\]
so \(\pi\) preserves the row Cartan matrix. One-dimensional root spaces give nonzero scalars \(c_i\) with \(\sigma e_i=c_i e_{\pi(i)}\). The bracket relation forces
\(\sigma f_i=c_i^{-1}f_{\pi(i)}\). Since \(\sigma\) commutes with \(\tau\), it is unitary for \(B\). The normalization
\[
B(e_i,e_i)=\tfrac12K(h_i,h_i)
=\tfrac12K(h_{\pi(i)},h_{\pi(i)})
=B(e_{\pi(i)},e_{\pi(i)})
\]
therefore gives \(|c_i|=1\). Also \(\sigma^2=1\) gives \(c_i c_{\pi(i)}=1\); at fixed vertices this means \(c_i=\pm1\).

For each two-element orbit \(i,j=\pi(i)\), replace
\[
e_i\ \text{by }e_i,\quad e_j\ \text{by }c_i e_j,
\qquad
f_i\ \text{by }f_i,\quad f_j\ \text{by }c_i^{-1}f_j.
\tag{AB.7}
\]
These are the inverse scalings of Z.3, so preserve all presentation relations. Since \(|c_i|=1\), they also preserve \(\tau e_j=-f_j\). The two new coefficients are \(c_i/c_i=1\) and \(c_ic_j=1\). This gives exactly (AB.5).

For the converse, start with any finite-type row Cartan matrix and its algebra from U.1. A permutation \(\pi\) preserving the matrix takes every Serre relation to the identically indexed relation. The extra inverse scalings by \(c_i\) likewise preserve every relation, as checked in Z.3. Thus (AB.5) defines a bracket-preserving linear map, and applying it twice fixes every generator because \(\pi^2=1\) and \(c_ic_{\pi(i)}=1\). It is an involutive automorphism. Its coefficients here are real signs, so direct substitution in (AA.2) proves that it commutes with \(\tau\), and hence restricts to the compact real form.

For a general compact real form with its adapted generators, the U.1 presentation isomorphism intertwines its \(\tau\) with (AA.2). It is therefore an isomorphism on the compact real fixed algebras as well. Formula (AB.5) transports the original involution to one of the listed representatives. For a fixed \(\mathfrak u\), any other identification differs by an automorphism of \(\mathfrak u\); consequently this is the asserted surjectivity onto conjugacy classes, without any uniqueness claim. □

## AC. Matrix foundations for the classical involutions

We use \(\mathfrak{sp}(n)\) for the compact algebra of rank \(n\), and \(\mathfrak{sp}_{2n}(\mathbb C)\) or \(\mathfrak{sp}_{2n}(\mathbb R)\) for the algebras preserving a nondegenerate alternating form on a \(2n\)-dimensional vector space over the indicated field. A superscript \(T\) means transpose and \(X^*=\bar X^T\). Put
\[
J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}.
\tag{AC.1}
\]

**Theorem AC.1 (the classical compact forms and their complete root decompositions).** The following compact real algebras have the indicated complexifications, roots and real dimensions:

| Compact algebra | Complexification | Roots | Dimension |
| --- | --- | --- | --- |
| \(\mathfrak{su}(N)\), \(N\geq2\) | \(\mathfrak{sl}_N(\mathbb C)\) | \(\varepsilon_i-\varepsilon_j,\ i\ne j\), on \(\sum t_i=0\) | \(N^2-1\) |
| \(\mathfrak{so}(2n+1)\), \(n\geq1\) | \(\mathfrak{so}_{2n+1}(\mathbb C)\) | \(\pm\varepsilon_i\pm\varepsilon_j\ (i<j),\ \pm\varepsilon_i\) | \(n(2n+1)\) |
| \(\mathfrak{sp}(n)\), \(n\geq1\) | \(\mathfrak{sp}_{2n}(\mathbb C)\) | \(\pm\varepsilon_i\pm\varepsilon_j\ (i<j),\ \pm2\varepsilon_i\) | \(n(2n+1)\) |
| \(\mathfrak{so}(2n)\), \(n\geq2\) | \(\mathfrak{so}_{2n}(\mathbb C)\) | \(\pm\varepsilon_i\pm\varepsilon_j\ (i<j)\) | \(n(2n-1)\) |

Here \(\mathfrak{su}(N)=\{X:X^*=-X,\ \operatorname{tr}X=0\}\),
\(\mathfrak{sp}(n)=\{X:X^*=-X,\ X^TJ_n+J_nX=0\}\), and \(\mathfrak{so}(M)\) consists of real skew-symmetric \(M\)-by-\(M\) matrices. Each listed algebra is semisimple and has negative-definite Killing form. The types are \(A_{N-1},B_n,C_n,D_n\), respectively, with \(B_1=C_1=A_1\), \(D_2=A_1\oplus A_1\), and \(D_3=A_3\). Every root space is one dimensional. The commutant of each complex algebra on its displayed defining vector space consists only of scalar matrices.

**Proof.** For \(\mathfrak{sl}_N\), take trace-zero diagonal matrices \(H=\operatorname{diag}(t_1,\ldots,t_N)\). The matrix unit \(E_{ij}\), \(i\ne j\), has weight \(t_i-t_j\), since \([H,E_{ij}]=(t_i-t_j)E_{ij}\). The zero-weight subspace is exactly the diagonal trace-zero space. These matrices are a basis, giving all weights, their multiplicities and the dimension.

For the symplectic case, solving \(X^TJ_n+J_nX=0\) entry by entry gives
\[
X=\begin{pmatrix}A&B\\C&-A^T\end{pmatrix},
\qquad B^T=B,\quad C^T=C.
\tag{AC.2}
\]
Take \(H=\operatorname{diag}(t_1,\ldots,t_n,-t_1,\ldots,-t_n)\). The off-diagonal entries of \(A\) give the vectors
\(E_{ij}-E_{n+j,n+i}\) of weight \(\varepsilon_i-\varepsilon_j\). For \(i<j\), the entries of \(B\) give
\(E_{i,n+j}+E_{j,n+i}\) of weight \(\varepsilon_i+\varepsilon_j\); its diagonal entries give \(E_{i,n+i}\) of weight \(2\varepsilon_i\). The entries of \(C\) give the corresponding negative weights. Diagonal \(A\) is the entire zero-weight space. This proves the complete root list and the dimension
\(n^2+2n(n+1)/2=n(2n+1)\).

For the even orthogonal case, choose a complex basis in which the symmetric bilinear form has matrix
\[
S=\begin{pmatrix}0&I_n\\I_n&0\end{pmatrix}.
\]
Such a basis is obtained from real orthonormal pairs \(a_i,b_i\) by
\(z_i=(a_i+ib_i)/\sqrt2,\ w_i=(a_i-ib_i)/\sqrt2\).
It is also unitary for the usual positive Hermitian form. The equation \(X^TS+SX=0\) again gives (AC.2), but now \(B^T=-B,\ C^T=-C\). Replace the plus signs in the off-diagonal \(B,C\) root vectors by minus signs, and omit their diagonal entries. Thus the only roots are \(\pm\varepsilon_i\pm\varepsilon_j\), \(i<j\), and the dimension is
\(n^2+n(n-1)=n(2n-1)\).

For the odd orthogonal case add a real unit vector \(v_0\), orthogonal to the preceding pairs. With the ordered coordinates \(z,w,v_0\), the form is \(\operatorname{diag}(S,1)\), and solving its infinitesimal preservation equation gives
\[
X=\begin{pmatrix}
A&B&u\\ C&-A^T&v\\-v^T&-u^T&0
\end{pmatrix},\qquad B^T=-B,\quad C^T=-C.
\tag{AC.3}
\]
The previous weights persist. The vector \(u_i\) contributes
\(E_{i,0}-E_{0,n+i}\) of weight \(\varepsilon_i\), and \(v_i\) contributes
\(E_{n+i,0}-E_{0,i}\) of weight \(-\varepsilon_i\); the index \(0\) here denotes the last coordinate. There are no other entries. The dimension is \(n(2n-1)+2n=n(2n+1)\). For \(n=1\), only these last two root spaces occur.

The map \(\tau X=-X^*\) preserves each displayed complex algebra and is a conjugation. For (AC.2) this follows by transposing the symmetric block conditions. For the orthogonal models, the preceding unitary change of basis reduces the assertion to real skew-symmetric matrices, where \(-X^*=\bar X\). Its fixed algebras are precisely the compact algebras in the table. Formula (AA.1) proves that their complexifications are the full displayed complex algebras.

Before using semisimple root theory, we can verify semisimplicity directly. The real form
\[
Q(X,Y)=-\operatorname{Re}\operatorname{tr}(XY)
\tag{AC.4}
\]
is positive definite on skew-Hermitian matrices and invariant under brackets by cyclicity of trace. The centre of each complex algebra is zero: a central element must lie in the zero-weight diagonal space, and the listed roots span its dual, forcing every diagonal coordinate to vanish. For \(D_2\), the two independent roots \(\varepsilon_1-\varepsilon_2,\varepsilon_1+\varepsilon_2\) still span. Thus the real centre is also zero. Every adjoint operator on the compact algebra is skew for \(Q\), and for an orthonormal real basis \(v_j\),
\[
K(X,X)=-\sum_j\|[X,v_j]\|_Q^2<0\quad(X\ne0).
\tag{AC.5}
\]
[Symmetric spaces H.5–H.6](symmetric-spaces.md#theorem-h-5) prove semisimplicity over both fields.

The displayed diagonal algebra is toral and equals its centralizer, so is a Cartan. The Killing form on it is the sum of products of root values. In the four cases it is, respectively,
\[
2N\sum_i t_i s_i,\quad
2(2n-1)\sum_i t_i s_i,\quad
4(n+1)\sum_i t_i s_i,\quad
4(n-1)\sum_i t_i s_i.
\tag{AC.6}
\]
For \(A\), expand \(\sum_{i\ne j}(t_i-t_j)(s_i-s_j)\) and use both zero sums. For the other types, the four roots for each pair \(i<j\) cancel mixed products and contribute \(4t_is_i+4t_js_j\). The extra roots contribute \(2t_is_i\) for \(B\) and \(8t_is_i\) for \(C\). Thus their canonical root metrics are positive multiples of the ordinary coordinate metrics. The complete coordinate simple-basis and Cartan calculations in [De Rham decomposition AE.3](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-3), with the row convention of P.1, identify the types. The rank-one identifications are immediate from a single opposite pair. \(D_2\) is the orthogonal union of its two opposite pairs. For \(D_3\), the basis
\(\varepsilon_1-\varepsilon_2,\ \varepsilon_2-\varepsilon_3,\ \varepsilon_2+\varepsilon_3\)
has a chain diagram with three vertices, hence type \(A_3\). U.1 then gives the asserted Lie-algebra identifications. A connected diagram gives simplicity by T.4; \(D_2\) has its two simple summands.

Finally let a matrix \(M\) commute with the complex algebra. Commuting with the diagonal Cartan makes \(M\) diagonal, because all its weights on the defining vector space are distinct linear functionals: \(t_i\) for \(A\), the \(t_i,-t_i\) for \(C,D\), and these together with zero for \(B\). The finite-hyperplane argument in [De Rham decomposition AD.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ad-5) gives a Cartan element separating these finitely many weights, so the assertion follows entry by entry. A nonzero entry of any root matrix then forces the two corresponding diagonal entries of \(M\) to agree. The \(A\) matrix units connect every pair. In \(C\), the \(A\) entries connect positive coordinates to one another and negative coordinates to one another, and \(E_{i,n+i}\) joins the two sets. In \(D\), the \(B\) entries join them for \(n\geq2\). In \(B\), the short-root entries join both sets through the last coordinate, also when \(n=1\). Thus all entries agree and \(M\) is scalar. □

**Lemma AC.2 (real, quaternionic and symplectic linear algebra).** An antiunitary map \(j\) on a finite-dimensional complex Hermitian space with \(j^2=1\) is conjugate by a unitary basis change to ordinary conjugation. If \(j^2=-1\), the dimension is even, and in a unitary basis \(j=J_n\kappa\), where \(\kappa\) is entrywise conjugation. The complex-linear operators commuting with \(J_n\kappa\) are exactly
\[
\Phi(A+B\mathbf j)=
\begin{pmatrix}A&B\\-\bar B&\bar A\end{pmatrix},
\tag{AC.7}
\]
the matrix realization of quaternionic matrices. It preserves products and adjoints. In particular,
\[
\mathfrak{sp}(n)=\{Q\in M_n(\mathbb H):Q^\dagger=-Q\},
\tag{AC.8}
\]
and the complex trace in (AC.7) is \(2\operatorname{Re}\operatorname{tr}_{\mathbb H}Q\).

**Proof.** If \(j^2=1\), formula (AA.1), applied to the vector space, writes it as \(V_{\mathbb R}+iV_{\mathbb R}\), where \(V_{\mathbb R}\) is the fixed real space. The Hermitian product is real on \(V_{\mathbb R}\), since antiunitarity conjugates it. Real Gram–Schmidt gives an orthonormal real basis there; it is a complex basis of the whole space, and \(j\) is conjugation in that basis.

If \(j^2=-1\), a unit vector \(v\) is orthogonal to \(jv\). Indeed antiunitarity applied to \(v,jv\) gives
\(\langle jv,j^2v\rangle=\overline{\langle v,jv\rangle}\), whereas \(j^2v=-v\) makes the left side its negative. The complex span of \(v,jv\) is \(j\)-stable. Its orthogonal complement is also \(j\)-stable, by antiunitarity. Induction produces orthonormal pairs; choose the order and sign in each pair so that the matrix is \(J_n\kappa\). This also proves even dimension. Any two such paired orthonormal bases are related by a unitary operator commuting with \(j\).

For completeness, define \(\mathbb H\) as the real algebra of pairs \(a+b\mathbf j\), \(a,b\in\mathbb C\), with multiplication
\[
(a+b\mathbf j)(c+d\mathbf j)
=(ac-b\bar d)+(ad+b\bar c)\mathbf j.
\tag{AC.9}
\]
The injective map
\[
a+b\mathbf j\longmapsto
\begin{pmatrix}a&b\\-\bar b&\bar a\end{pmatrix}
\]
respects multiplication by direct multiplication of its four entries. It proves associativity. It also gives \(\mathbf j^2=-1,\ \mathbf j a=\bar a\mathbf j\), and conjugation
\(\overline{a+b\mathbf j}=\bar a-b\mathbf j\) corresponds to conjugate transpose. The product with this conjugate is \((|a|^2+|b|^2)I\), so every nonzero quaternion is invertible. Applying the entry formula to matrices proves (AC.7), product preservation and adjoint preservation; here \(\dagger\) means quaternionic conjugate transpose. Solving \(XJ_n=J_n\bar X\) in four blocks gives exactly (AC.7).

For \(X\in\mathfrak{sp}_{2n}(\mathbb C)\), the defining equation gives
\[
J_n\bar XJ_n^{-1}=-X^*.
\tag{AC.10}
\]
Thus within that complex algebra, commuting with \(J_n\kappa\) is equivalent to being skew-Hermitian. Conversely a skew-Hermitian matrix commuting with \(J_n\kappa\) satisfies the symplectic equation by reversing this calculation. This proves (AC.8). Taking diagonal traces in (AC.7) proves the final trace assertion.

We record two elementary basis constructions that will be used below. A real orthogonal operator with square \(1\) has orthogonal \(+1,-1\) eigenspaces: its inverse equals itself, so it is self-adjoint and the real spectral theorem, [De Rham decomposition H.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-h-2), applies. A real orthogonal operator with square \(-1\) is skew-adjoint; a real unit vector \(v\) and its image are orthogonal, and the orthogonal complement of their plane is invariant. Induction gives the usual two-dimensional rotation blocks. Finally a nondegenerate alternating real bilinear form admits a symplectic basis: choose \(v,w\) with their pairing one; subtract their two pairing components from any vector to obtain the orthogonal complement of their plane for the alternating form. This is a direct sum, its complementary restriction is nondegenerate, and induction gives all the pairs. These constructions prove the asserted normal forms without a classification theorem about forms. □

**Lemma AC.3 (toral phases and compact reflection representatives).** Fix the normalized generators and compact conjugation of AA.1. Every automorphism fixing the Cartan pointwise and preserving the compact real form is
\(\exp(\operatorname{ad}T)\) for some \(T\in\mathfrak t=\mathfrak h\cap\mathfrak u\).
Every compact involution is conjugate under \(\operatorname{Aut}_{\mathbb R}(\mathfrak u)\) to the product of such a toral automorphism and a diagram permutation of order at most two.
Each simple reflection is induced on \(\mathfrak h\) by the compact inner automorphism
\[
n_i=\exp\bigl(\tfrac{\pi}{2}\operatorname{ad}(e_i-f_i)\bigr).
\tag{AC.11}
\]
For a toral involution with generator signs \(c_j=\pm1\), conjugation by \(n_i\) changes them to
\[
c'_j=c_jc_i^{-a_{ij}}.
\tag{AC.12}
\]

**Proof.** A Cartan-pointwise automorphism acts on the one-dimensional simple root spaces by \(e_i\mapsto c_ie_i\). The bracket relation gives \(f_i\mapsto c_i^{-1}f_i\), and compactness preservation gives \(|c_i|=1\), as in AB.2. Write \(c_i=e^{i\theta_i}\). The simple roots are a basis of \(\mathfrak h_{\mathbb R}^*\), where \(\mathfrak h_{\mathbb R}\) is the real span of the \(h_i\), by Q.5. Solve \(\alpha_i(H)=\theta_i\) there and put \(T=iH\in\mathfrak t\). Then \(\exp(\operatorname{ad}T)\) has exactly these actions on all generators. Since the generators generate \(\mathfrak g\), the automorphisms agree. The elementary unit-circle parameterization used here is [Connections E.1](connections-and-parallel-transport.md#lemma-e-1).

AB.2 supplies adapted normalized generators for any compact involution. U.1 identifies this presentation with the fixed one, intertwining the compact conjugations because their values on the generators agree. The resulting identification restricts to a compact real automorphism. In these coordinates, remove the pure diagram permutation from the involution; the remainder fixes \(\mathfrak h\) pointwise and is the toral automorphism just proved. This proves the product assertion; it does not assert a unique product or unique conjugacy representative. More generally any other compact automorphism inducing the same permutation of these simple roots can replace the pure diagram map: their quotient fixes the Cartan pointwise, so their difference is again a toral factor.

The vector \(e_i-f_i\) lies in \(\mathfrak u\). For any real derivation \(D\), differentiating \(e^{-tD}[e^{tD}X,e^{tD}Y]\) gives zero by the derivation rule. The finite-dimensional linear ODE and its exponential solution are [Local tools 2.3](local-tools-for-bundles-and-transport.md#2-differential-equations-and-their-parameters). Hence its adjoint exponential preserves brackets and \(\mathfrak u\). On the span of \(h_i,e_i+f_i\), its adjoint sends
\[
h_i\longmapsto-2(e_i+f_i),\qquad
e_i+f_i\longmapsto2h_i.
\]
The exponential at time \(\pi/2\) therefore sends \(h_i\) to \(-h_i\). If \(H\in\ker\alpha_i\), it commutes with \(e_i,f_i\) and is fixed. Decomposing \(H\) into this kernel and \(\alpha_i(H)h_i/2\) proves the reflection formula \(H\mapsto H-\alpha_i(H)h_i\). Products of the \(n_i\) consequently realize every element of the Weyl group.

The action of a toral automorphism on a root
\(\alpha=\sum_jm_j\alpha_j\) is the scalar \(\prod_jc_j^{m_j}\): this follows either from its exponential formula or from the generating root brackets in U.1. Conjugating by \(n_i\) replaces \(\alpha_j\) by
\(s_i\alpha_j=\alpha_j-a_{ij}\alpha_i\), which gives (AC.12). In particular, when \(c_i=-1\), exactly the neighbours with odd row entry \(a_{ij}\) change sign. This is a proved conjugacy operation; no assertion that these operations give all equivalences is needed here. □

## AD. Classification of the classical compact involutions

For a compact involution \(\sigma\), write \(\mathfrak k=\mathfrak u_+\), \(\mathfrak p=\mathfrak u_-\), and call \(\mathfrak k+i\mathfrak p\) its real dual. This is the real form proved in AA.5. In the following statements, conjugacy means conjugacy under all real automorphisms of the compact algebra.

**Theorem AD.1 (unitary algebras).** Every involution of \(\mathfrak{su}(N)\), \(N\geq2\), is conjugate to one of the following, including the identity case in the first row:

| Kind | Fixed algebra \(\mathfrak k\) | Real dual | \(\dim_{\mathbb R}\mathfrak p\) |
| --- | --- | --- | --- |
| Unitary signs, \(p+q=N\) | \(\mathfrak s(\mathfrak u(p)\oplus\mathfrak u(q))\) | \(\mathfrak{su}(p,q)\) | \(2pq\) |
| Real structure | \(\mathfrak{so}(N)\) | \(\mathfrak{sl}_N(\mathbb R)\) | \((N-1)(N+2)/2\) |
| Quaternionic structure, \(N=2m\) | \(\mathfrak{sp}(m)\) | \(\mathfrak{su}^*(2m)\) | \((m-1)(2m+1)\) |

The representatives in row order are \(\operatorname{Ad}\operatorname{diag}(I_p,-I_q)\),
\(X\mapsto\bar X\), and \(X\mapsto J_m\bar XJ_m^{-1}\) on \(\mathfrak{su}(N)\).
Here \(\mathfrak s\) imposes zero total complex trace, \(\mathfrak{su}^*(2m)=\mathfrak{sl}_m(\mathbb H)\), and
\(\mathfrak{sl}_m(\mathbb H)\) means quaternionic matrices with zero real part of the quaternionic trace. In the first row \(p,q\geq0\), and interchanging them does not change the involution class. For \(N=2\), the second row repeats the first row with \(p=q=1\), and the third row is the identity. For \(N\geq3\), the displayed nonidentity rows, with unordered \(p,q\), are distinct.

**Proof.** The \(A_{N-1}\) diagram is a chain. For \(N\geq3\) its only permutations preserving the Cartan matrix are the identity and reversal; for \(N=2\) there is only the identity. Use the standard generators \(e_i=E_{i,i+1}, f_i=E_{i+1,i}, h_i=E_{ii}-E_{i+1,i+1}\). The map \(\gamma(X)=-X^T\) preserves \(\mathfrak{sl}_N\), since transposition reverses products; on its compact real form it is \(X\mapsto\bar X\). Let \(R\) be the real antidiagonal matrix with antidiagonal entries \(R_{i,N+1-i}=(-1)^{i-1}\). It is unitary, \(R^2=(-1)^{N-1}I\), and direct substitution gives
\[
\operatorname{Ad}R\circ\gamma:
\quad e_i\mapsto e_{N-i},\quad
f_i\mapsto f_{N-i},\quad h_i\mapsto h_{N-i}.
\tag{AD.1}
\]
For instance conjugation of \(E_{i+1,i}\) by \(R\) gives \(-E_{N-i,N+1-i}\), canceling the minus sign in \(\gamma\). Thus (AD.1) realizes diagram reversal.

By AC.3, every involution is conjugate to \(\operatorname{Ad}U\) or \(\operatorname{Ad}U\circ\gamma\) for a unitary matrix \(U\): the toral factor is conjugation by the diagonal unitary matrix \(\exp T\), and (AD.1) gives the other possibility. We now classify these two kinds directly.

For \(\sigma=\operatorname{Ad}U\), its square is the identity precisely when \(U^2\) commutes with \(\mathfrak{sl}_N\). By AC.1, \(U^2=\lambda I\), with \(|\lambda|=1\). Multiply \(U\) by a unit scalar whose square is \(\lambda^{-1}\); this does not change \(\operatorname{Ad}U\) and makes \(U^2=I\). Such a unitary is self-adjoint. K.1 gives orthogonal \(+1,-1\) eigenspaces of dimensions \(p,q\), so unitary conjugation makes \(U=\operatorname{diag}(I_p,-I_q)\). Replacing \(U\) by \(-U\) interchanges the dimensions and leaves its adjoint action unchanged.

The even part consists of block-diagonal skew-Hermitian matrices of zero total trace. The odd part consists of
\[
X_Z=\begin{pmatrix}0&Z\\-Z^*&0\end{pmatrix},
\qquad Z\in M_{p,q}(\mathbb C).
\tag{AD.2}
\]
Multiplication by \(i\) in the dual replaces these by all blocks
\(\begin{psmallmatrix}0&W\\W^*&0\end{psmallmatrix}\), by putting \(W=iZ\). Together with the same even part, these are exactly the trace-zero matrices satisfying
\[
X^*S+SX=0,\qquad S=\operatorname{diag}(I_p,-I_q).
\]
This defines \(\mathfrak{su}(p,q)\). The odd real dimension is \(2pq\).

For \(\sigma=\operatorname{Ad}U\circ\gamma\), the formula on \(\mathfrak{su}(N)\) is \(X\mapsto U\bar XU^{-1}\). Let \(j=U\kappa\), an antiunitary operator on \(\mathbb C^N\). The condition \(\sigma^2=1\) and AC.1 give \(j^2=U\bar U=\lambda I\), where \(|\lambda|=1\). Conjugating this equality entrywise gives \(\bar U U=\bar\lambda I\); conjugating the latter by \(U\) gives \(U\bar U=\bar\lambda I\), so \(\lambda=\bar\lambda\) and \(\lambda=\pm1\).

If \(j^2=1\), AC.2 puts \(j=\kappa\) in a unitary basis. Then \(\mathfrak k\) consists of real skew-symmetric matrices, namely \(\mathfrak{so}(N)\). On all of \(\mathfrak{sl}_N(\mathbb C)\), the compact conjugation is \(\tau X=-X^*\), while the complex extension of \(\sigma\) is \(-X^T\). Their product is entrywise conjugation. Thus the real dual is \(\mathfrak{sl}_N(\mathbb R)\), with odd dimension
\[
(N^2-1)-N(N-1)/2=(N-1)(N+2)/2.
\]
If \(j^2=-1\), AC.2 gives \(N=2m\) and \(j=J_m\kappa\). Its commuting skew-Hermitian matrices are \(\mathfrak{sp}(m)\). The product \(\sigma\tau\) on the complex algebra is \(X\mapsto J_m\bar XJ_m^{-1}\), so its fixed matrices are (AC.7) with complex trace zero, equivalently zero real quaternionic trace. This is the stated \(\mathfrak{sl}_m(\mathbb H)\), also denoted \(\mathfrak{su}^*(2m)\). Its odd dimension is
\[
(4m^2-1)-m(2m+1)=(m-1)(2m+1).
\]

These arguments prove exhaustion. For \(N\geq3\), the nontrivial first-row fixed algebra has one-dimensional centre: its central elements are
\(\operatorname{diag}(iaI_p,ibI_q)\) with \(pa+qb=0\). This follows by commuting with the matrix units in each block; it remains valid if a block has size one. The other two fixed algebras have zero centre by AC.1, so cannot be conjugate to this row. When both the second and third rows occur, their fixed dimensions are respectively \(m(2m-1)\) and \(m(2m+1)\), which are different. Within the first row, the odd dimension \(2p(N-p)\) determines the unordered pair \(p,q\); it is strictly increasing for \(0\leq p\leq N/2\), since
\((p+1)(N-p-1)-p(N-p)=N-2p-1>0\) before the last distinct value. The identity has zero odd part and is separate.

For \(N=2\), direct multiplication on \(e,f,h\) gives \(\gamma=\operatorname{Ad}J_1\). Multiplication of \(J_1\) by \(i\) gives a Hermitian unitary with eigenvalues \(1,-1\), so this is the first row with \(p=q=1\). Composing once more with \(\operatorname{Ad}J_1\) gives the identity, which is the third row at \(m=1\). This proves the boundary identifications. □

**Theorem AD.2 (symplectic algebras).** Every involution of \(\mathfrak{sp}(n)\) is conjugate to one of the following:

| Kind | Fixed algebra | Real dual | Odd dimension |
| --- | --- | --- | --- |
| Quaternionic signs, \(p+q=n\) | \(\mathfrak{sp}(p)\oplus\mathfrak{sp}(q)\) | \(\mathfrak{sp}(p,q)\) | \(4pq\) |
| Complex structure | \(\mathfrak u(n)\) | \(\mathfrak{sp}_{2n}(\mathbb R)\) | \(n(n+1)\) |

The first representative is conjugation by \(+1\) on \(p\) quaternionic coordinates and \(-1\) on \(q\); the second is \(\operatorname{Ad}\operatorname{diag}(iI_n,-iI_n)\) in the complex realization. In the first row \(p,q\geq0\), their interchange is redundant, and a zero parameter gives the identity. The notation \(\mathfrak{sp}(p,q)\) means quaternionic matrices satisfying
\[
X^\dagger S+SX=0,\qquad S=\operatorname{diag}(I_p,-I_q).
\tag{AD.3}
\]
The two nonidentity kinds are distinct, and the unordered pair \(p,q\) distinguishes the first-row classes.

**Proof.** The \(C_n\) diagram has no nonidentity Cartan-matrix permutation: its end arrow distinguishes the two ends when \(n\geq2\), and for \(n=1\) there is one vertex. Thus AC.3 makes every involution conjugate to \(\operatorname{Ad}U\), where \(U=\exp T\) is both unitary and complex symplectic. It commutes with \(j=J_n\kappa\). Its square commutes with the defining complex algebra, so AC.1 gives \(U^2=\lambda I\). Since \(U^2\) preserves \(J_n\), one has \(\lambda^2=1\).

If \(U^2=1\), its orthogonal complex \(+1,-1\) eigenspaces are \(j\)-stable, because \(Uj=jU\). By AC.2 they have even dimensions \(2p,2q\), and each has a paired orthonormal basis. Joining these bases is a unitary change commuting with \(j\), hence symplectic: for a unitary \(V\), the equation \(VJ_n=J_n\bar V\) is equivalent to \(V^TJ_nV=J_n\), by multiplying and using \(V^*V=I\). It is therefore an allowed compact matrix conjugation. In quaternionic coordinates, \(U\) is now \(\operatorname{diag}(I_p,-I_q)\). Its fixed skew-Hermitian quaternionic matrices are the two diagonal \(\mathfrak{sp}\) blocks. Its odd part consists of
\[
X_B=\begin{pmatrix}0&B\\-B^\dagger&0\end{pmatrix},
\qquad B\in M_{p,q}(\mathbb H).
\tag{AD.4}
\]

To identify the dual without confusing the complexification scalar \(i\) with quaternionic left multiplication, use the real linear map
\[
\begin{pmatrix}A&0\\0&D\end{pmatrix}+iX_B
\longmapsto
\begin{pmatrix}A&B\\B^\dagger&D\end{pmatrix}.
\tag{AD.5}
\]
Its target is exactly (AD.3), with \(A^\dagger=-A,\ D^\dagger=-D\). It preserves even-even brackets. Both even-odd brackets send \(B\) to \(AB-BD\). Finally
\[
[X_B,X_C]=
\operatorname{diag}(-BC^\dagger+CB^\dagger,\,
                   -B^\dagger C+C^\dagger B).
\tag{AD.6}
\]
The bracket of the two positive-lower-block matrices on the right of (AD.5) is the negative of (AD.6), exactly the sign supplied by \(i^2=-1\) on the left. Thus (AD.5) is an isomorphism of real Lie algebras. The odd dimension is \(4pq\).

If \(U^2=-1\), its eigenvalues are \(i,-i\). Since \(j\) is antilinear and commutes with \(U\), it exchanges these eigenspaces, which are orthogonal by unitary diagonalization from K.1. Choose an orthonormal basis in the \(i\)-eigenspace and its paired images under \(j\). Both eigenspaces have dimension \(n\), and the resulting compact symplectic basis change puts \(U=\operatorname{diag}(iI_n,-iI_n)\). In (AC.2), fixedness forces \(B=C=0\); the compact condition then says \(A^*=-A\). Therefore \(\mathfrak k\simeq\mathfrak u(n)\).

For the real dual, (AC.10) identifies \(\tau\) on the complex algebra with conjugation by the antiunitary \(j\). Hence \(\sigma\tau\) is conjugation by \(c=Uj\). Its square on vectors is
\(c^2=U^2j^2=1\). Let \(W\) be the real fixed space of \(c\). By AC.2 its complexification is the defining \(\mathbb C^{2n}\). The complex alternating form \(\beta(v,w)=v^TJ_nw\) satisfies
\[
\beta(jv,jw)=\overline{\beta(v,w)},\qquad
\beta(cv,cw)=\overline{\beta(v,w)},
\]
the first by \(J_n^TJ_n=I\), and the second because \(U\) is symplectic. Thus \(\beta|_W\) is real. It is nondegenerate: a vector of \(W\) annihilating \(W\) also annihilates \(W+iW\), hence vanishes. The real symplectic-basis construction in AC.2 identifies its infinitesimal preservers with \(\mathfrak{sp}_{2n}(\mathbb R)\). These are precisely the complex symplectic matrices commuting with \(c\), namely the real dual. Subtracting fixed dimensions gives \(n(2n+1)-n^2=n(n+1)\).

The nontrivial first-row fixed algebra is centre-free by AC.1, whereas \(\mathfrak u(n)\) has centre \(i\mathbb RI_n\), so the rows are distinct. Within the first row, \(4p(n-p)\) determines the unordered pair by the same strict-increase calculation as in AD.1. A quaternionic coordinate permutation interchanges the blocks, and replacing \(U\) by \(-U\) leaves its adjoint action unchanged. The identity is exactly the zero-odd-part case. □

**Theorem AD.3 (orthogonal algebras).** For \(N\geq3\), every involution of \(\mathfrak{so}(N)\), up to all compact-algebra automorphisms, has one of the following representatives:

| Kind | Fixed algebra | Real dual | Odd dimension |
| --- | --- | --- | --- |
| Orthogonal signs, \(p+q=N\) | \(\mathfrak{so}(p)\oplus\mathfrak{so}(q)\) | \(\mathfrak{so}(p,q)\) | \(pq\) |
| Complex structure, \(N=2n\) | \(\mathfrak u(n)\) | \(\mathfrak{so}^*(2n)\) | \(n(n-1)\) |

The representatives are respectively \(\operatorname{Ad}\operatorname{diag}(I_p,-I_q)\) and \(\operatorname{Ad}J_n\). The first row permits zero parameters and is unchanged by interchanging them. We define the last real form explicitly by
\[
\mathfrak{so}^*(2n)
=\{X\in M_{2n}(\mathbb C):X^T+X=0,\ XJ_n=J_n\bar X\}.
\tag{AD.7}
\]
It is also the algebra preserving a nondegenerate quaternionic skew-Hermitian form on \(\mathbb H^n\). The assertion is exhaustion by representatives; it permits repetitions under additional automorphisms in low rank.

**Proof.** First we show that every class has a representative \(\operatorname{Ad}U\) for a real orthogonal matrix \(U\). For odd \(N=2n+1\), the \(B_n\) diagram has no nonidentity permutation, including \(n=1\). AC.3 therefore gives \(U=\exp T\in SO(N)\).

For even \(N=2n\), use the root coordinates of AC.1. A real reflection sending \(b_n\) to \(-b_n\) and fixing the other real basis vectors exchanges \(z_n,w_n\), hence changes \(\varepsilon_n\) to \(-\varepsilon_n\). It swaps the simple roots \(\varepsilon_{n-1}-\varepsilon_n,\varepsilon_{n-1}+\varepsilon_n\) and fixes the other simple roots. This realizes the nontrivial diagram permutation for \(D_n\), \(n\geq5\), also for the chain \(D_3=A_3\), and for the two components of \(D_2\). For \(D_4\), its three equal arms give all permutations of three ends. An order-two diagram permutation is either the identity or a transposition. Conjugating it by a diagram automorphism makes it the transposition just realized. Such conjugation is an actual compact automorphism by the Serre generator permutations in AB.2; no matrix realization of this extra \(D_4\) automorphism is assumed.

After this possible \(D_4\) conjugation, AC.3 expresses the involution as a toral factor times the chosen real reflection \(R\), or as a toral factor alone. Although \(R\) may introduce phases on the chosen root vectors, the final assertion in the second paragraph of the proof of AC.3 absorbs those phases into the toral factor. Thus the representative is \(\operatorname{Ad}U\), with \(U=\exp T\,R\) or \(U=\exp T\), a real orthogonal matrix. This proves the claim for all the stated ranks without asserting that every \(D_4\) automorphism is a matrix conjugation.

Now \((\operatorname{Ad}U)^2=1\) and AC.1 imply \(U^2=\lambda I\). Reality and orthogonality give \(\lambda=\pm1\). The negative sign is impossible in odd dimension, since \(\det(U^2)=1\) whereas \(\det(-I)=-1\).

If \(U^2=1\), the real orthogonal basis construction in AC.2 puts it in the first-row form. The even part consists of the two skew-symmetric diagonal blocks; the odd part is (AD.4) with real \(B\) and \(\dagger=T\). The real version of (AD.5) and its complete bracket calculation (AD.6) identify the dual with the matrices satisfying
\(X^TS+SX=0\), where \(S=\operatorname{diag}(I_p,-I_q)\). This defines \(\mathfrak{so}(p,q)\), and the odd dimension is \(pq\). Replacing \(U\) by \(-U\) and permuting coordinates interchanges \(p,q\).

If \(U^2=-1\), the real two-plane construction in AC.2 gives \(N=2n\) and an orthogonal change of basis making \(U=J_n\). Its real commuting operators are exactly the complex-linear operators for the complex structure \(J_n\) on \(\mathbb R^{2n}\). In real coordinates, a complex matrix \(A+iB\) acts by
\(\begin{psmallmatrix}A&-B\\B&A\end{psmallmatrix}\); skew-symmetry is equivalent to \(A^T=-A,\ B^T=B\), or \((A+iB)^*=-(A+iB)\). Thus the fixed algebra is \(\mathfrak u(n)\). Compact conjugation on \(\mathfrak{so}_N(\mathbb C)\) in its original real coordinates is \(\tau X=\bar X\). Hence \(\sigma\tau X=J_n\bar XJ_n^{-1}\), whose fixed algebra is (AD.7). The odd dimension is
\[
n(2n-1)-n^2=n(n-1).
\]

We finally verify the quaternionic interpretation in the statement. On \(V=\mathbb C^{2n}\), set \(j=J_n\kappa\) and \(\beta(v,w)=v^Tw\). Give \(V\) the right quaternionic vector-space structure in which right multiplication by \(\mathbf j\) is \(j\); its compatibility with complex scalars is exactly the antilinearity of \(j\), and \(j^2=-1\) supplies the other relation in (AC.9). Put
\[
h(v,w)=\beta(jv,w),\qquad
\mathcal H(v,w)=h(v,w)-\mathbf j\,\beta(v,w).
\tag{AD.8}
\]
Here the \(\mathbf j\) is written on the left of the complex scalar. The identity \(\beta(jv,jw)=\overline{\beta(v,w)}\), together with \(j^2=-1\), shows that \(h\) is complex skew-Hermitian. Direct substitution gives
\[
\begin{aligned}
\mathcal H(vc,w)&=\bar c\,\mathcal H(v,w),&
\mathcal H(v,wc)&=\mathcal H(v,w)c\quad(c\in\mathbb C),\\
\mathcal H(jv,w)&=-\mathbf j\,\mathcal H(v,w),&
\mathcal H(v,jw)&=\mathcal H(v,w)\mathbf j,\\
\mathcal H(w,v)&=-\overline{\mathcal H(v,w)}.
\end{aligned}
\tag{AD.9}
\]
For example, \(h(v,jw)=\overline{\beta(v,w)}\) and
\(\beta(v,jw)=-\overline{h(v,w)}\), so the right-\(\mathbf j\) equation follows from
\(\mathbf j c=\bar c\mathbf j\). The left equation follows from
\(h(jv,w)=-\beta(v,w)\) and \(\beta(jv,w)=h(v,w)\).
The complex and \(\mathbf j\) equations imply the full quaternionic sesquilinearity by additivity and multiplication in (AC.9). Nondegeneracy follows from that of \(\beta\), by comparing the two complex components of \(\mathcal H\).

The first \(n\) standard complex vectors \(v_i\) form a quaternionic basis: together with \(jv_i=-v_{n+i}\) they form a complex basis. One has
\(\mathcal H(v_i,v_j)=-\mathbf j\,\delta_{ij}\); replacing every \(v_i\) by \(v_i i\) changes this to \(+\mathbf j\,\delta_{ij}\). Thus it is the usual diagonal nondegenerate quaternionic skew-Hermitian form. A complex-linear operator is quaternionic-linear exactly when it commutes with \(j\). For such an operator, infinitesimal preservation of \(\beta\) implies that of \(h\), hence of \(\mathcal H\). Conversely preservation of \(\mathcal H\) implies that of \(\beta\) by its two complex components. These conditions are exactly (AD.7). This proves the interpretation and finishes the exhaustion proof. □

## AE. Irreducible pairs and the classical Hermitian test

An involution of a compact semisimple algebra gives
\(\mathfrak u=\mathfrak k\oplus\mathfrak p\). The pair is **effective** if
\(\{X\in\mathfrak k:[X,\mathfrak p]=0\}=0\), and is **irreducible** if
\(\mathfrak p\ne0\) has no proper nonzero real subspace invariant under \(\mathfrak k\).

**Theorem AE.1 (simple and group-type irreducible pairs).** Every effective irreducible compact semisimple pair is of exactly one of the following structural kinds:

1. A compact simple algebra \(\mathfrak u\) with a nonidentity involution.
2. \(\mathfrak u=\mathfrak v\oplus\mathfrak v\), for a compact simple \(\mathfrak v\), with involution \((X,Y)\mapsto(Y,X)\).

Conversely every pair of either kind is effective and irreducible. The real dual of the second kind is
\((\mathfrak v_{\mathbb C},\mathfrak v)\), where \(\mathfrak v_{\mathbb C}\) is regarded as a real algebra. Compact simple algebras are precisely the compact real forms of the complex simple types listed in U.1.

**Proof.** By [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6), \(\mathfrak u\) is a direct sum of real simple ideals, and every ideal is a sum of those factors. Thus an automorphism permutes them. The orbits of an involution have size one or two; the sum over each orbit is a \(\sigma\)-stable ideal, and its odd part is a \(\mathfrak k\)-invariant subspace of \(\mathfrak p\). If an orbit had zero odd part, its ideal would lie entirely in \(\mathfrak k\) and commute with every other ideal, hence with all of \(\mathfrak p\). Effectiveness excludes it. Irreducibility therefore permits just one orbit.

A one-factor orbit gives the first kind, with a nonidentity involution because the odd part is nonzero. A two-factor orbit consists of isomorphic simple algebras. If \(\phi\) is the map from the first factor to the second, the other map is \(\phi^{-1}\), since \(\sigma^2=1\). Using \(\phi\) to identify the factors makes the involution the exchange, giving the second kind.

For the converse in a simple algebra, let \(Q=-K\). It is positive definite and invariant. Suppose \(\mathfrak p=V\oplus W\) with \(V\) a nonzero invariant subspace and \(W\) its \(Q\)-orthogonal complement. The complement is invariant because all \(\operatorname{ad}\mathfrak k\) are skew for \(Q\). For \(v\in V,w\in W,z\in\mathfrak k\),
\[
Q([v,w],z)=Q(v,[w,z])=0.
\]
Since \([v,w]\in\mathfrak k\), positivity gives \([V,W]=0\). Now
\(\mathfrak a=V+[V,V]\) is an ideal in \(\mathfrak u\). Indeed \(\mathfrak k\) preserves both summands by invariance and Jacobi; \([\mathfrak p,V]\subseteq[V,V]\); and \([\mathfrak p,[V,V]]\subseteq V\), using Jacobi, \([W,V]=0\), and the \(\mathfrak k\)-invariance of \(V\). Thus \(\mathfrak a\) is a nonzero ideal and equals \(\mathfrak u\). Its odd part is \(V\), so \(V=\mathfrak p\), proving irreducibility.

The ineffective kernel \(\mathfrak b\subseteq\mathfrak k\) is also an ideal in \(\mathfrak u\): Jacobi shows that \(\mathfrak k\) preserves it, and it commutes with \(\mathfrak p\) by definition. It is proper because \(\mathfrak p\ne0\), so simplicity makes it zero. This proves effectiveness.

For the exchange involution, the fixed and odd parts are
\[
\mathfrak k=\{(X,X)\},\qquad
\mathfrak p=\{(Y,-Y)\}.
\tag{AE.1}
\]
The isotropy action is the adjoint action of \(\mathfrak v\) on itself. Its invariant subspaces are ideals, so it is irreducible; its kernel is the centre, which is zero by [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6). The real linear map
\[
(X,X)+i(Y,-Y)\longmapsto X+iY
\tag{AE.2}
\]
identifies the dual with \(\mathfrak v_{\mathbb C}\) as a real algebra. To check brackets, the even-even and mixed terms are the same brackets in \(\mathfrak v\), while the odd-odd term on either side is \(-[Y,Y']\). The fixed algebra maps to \(\mathfrak v\).

It remains to justify the last classification assertion rather than assuming that a compact simple real algebra is absolutely simple. The complexification of a compact simple real algebra is semisimple by [Symmetric spaces H.6](symmetric-spaces.md#theorem-h-6). Its compact conjugation permutes its complex simple ideals. The sum over any orbit has a real fixed ideal, so real simplicity permits only one orbit. If that orbit had two factors, the real fixed algebra would be \(\{(Z,\eta Z)\}\) for an antilinear isomorphism \(\eta\), hence would be a complex simple algebra regarded as real. The Killing form there is \(2\operatorname{Re}K_{\mathbb C}(Z,Z)\). Nondegeneracy and polarization give some \(Z\) with \(K_{\mathbb C}(Z,Z)\ne0\); multiplying \(Z\) by a complex scalar makes this value positive real. Replacing \(Z\) by \(iZ\) reverses its sign. This contradicts compact negativity. Thus the complexification is simple.

Conversely a compact real form of a complex simple algebra is real simple: the complexification of a nonzero real ideal is a nonzero complex ideal, hence all of the complex algebra; comparison of dimensions then makes the real ideal all of the real form. U.1 classifies the complex simple algebras by connected diagrams. AA.1 gives a compact real form for each, and AA.3 proves uniqueness up to real isomorphism. This proves the assertion for all classical and exceptional types. □

**Corollary AE.2 (the classical algebraic list and its Hermitian entries).** The nontrivial classical pairs from AD.1–AD.3 have the following algebraic forms and odd dimensions:

| Family | Compact pair \((\mathfrak u,\mathfrak k)\) | Real dual algebra | \(\dim_{\mathbb R}\mathfrak p\) |
| --- | --- | --- | --- |
| AI, \(n\geq2\) | \(\begin{gathered}\mathfrak u=\mathfrak{su}(n)\\\mathfrak k=\mathfrak{so}(n)\end{gathered}\) | \(\mathfrak{sl}_n(\mathbb R)\) | \((n-1)(n+2)/2\) |
| AII, \(n\geq2\) | \(\begin{gathered}\mathfrak u=\mathfrak{su}(2n)\\\mathfrak k=\mathfrak{sp}(n)\end{gathered}\) | \(\mathfrak{su}^*(2n)\) | \((n-1)(2n+1)\) |
| AIII, \(p,q\geq1\) | \(\begin{gathered}\mathfrak u=\mathfrak{su}(p+q)\\\mathfrak k=\mathfrak s(\mathfrak u(p)\oplus\mathfrak u(q))\end{gathered}\) | \(\mathfrak{su}(p,q)\) | \(2pq\) |
| BDI, \(p,q\geq1\) | \(\begin{gathered}\mathfrak u=\mathfrak{so}(p+q)\\\mathfrak k=\mathfrak{so}(p)\oplus\mathfrak{so}(q)\end{gathered}\) | \(\mathfrak{so}(p,q)\) | \(pq\) |
| CI, \(n\geq1\) | \(\begin{gathered}\mathfrak u=\mathfrak{sp}(n)\\\mathfrak k=\mathfrak u(n)\end{gathered}\) | \(\mathfrak{sp}_{2n}(\mathbb R)\) | \(n(n+1)\) |
| CII, \(p,q\geq1\) | \(\begin{gathered}\mathfrak u=\mathfrak{sp}(p+q)\\\mathfrak k=\mathfrak{sp}(p)\oplus\mathfrak{sp}(q)\end{gathered}\) | \(\mathfrak{sp}(p,q)\) | \(4pq\) |
| DIII, \(n\geq3\) | \(\begin{gathered}\mathfrak u=\mathfrak{so}(2n)\\\mathfrak k=\mathfrak u(n)\end{gathered}\) | \(\mathfrak{so}^*(2n)\) | \(n(n-1)\) |

BDI with \((p,q)=(1,1)\) is flat, and BDI with \((p,q)=(2,2)\) is a product of two rank-one pairs, rather than irreducible. All other entries in the stated ranges are effective and irreducible. The nonflat irreducible Hermitian entries are AIII, CI, DIII, and BDI with one parameter equal to two, together with AI at \(n=2\), which repeats AIII at \(p=q=1\). CI at \(n=1\) repeats that rank-one pair as well. Each Hermitian entry has exactly the two invariant complex structures of E.2, interchanged by sign, for any positive scale of its invariant metric.

**Proof.** The seven algebraic rows and every dimension are precisely the three exhaustive calculations in AD.1–AD.3, with identity or ineffective small cases omitted by the indicated ranges. No matrix description is inferred merely from a family name. AC.1 shows that their compact algebras are simple except for the BDI cases with \(p+q=4\) and the explicitly flat \(p+q=2\). Consequently AE.1 proves effectiveness and irreducibility in all the simple cases.

For BDI at \((1,1)\), the compact algebra is one-dimensional abelian, its even part is zero, and every bracket vanishes. The curvature formula in D.1 therefore makes it flat. If \(p+q=4\), interchange parameters so the remaining possibilities are \((1,3)\) and \((2,2)\). For \((1,3)\), the odd matrices are a real three-dimensional row-vector space, and \(\mathfrak{so}(3)\) acts by its usual rotations. This representation is irreducible: for \(0\ne v\) and any \(w\perp v\), the skew matrix
\((wv^T-vw^T)/\|v\|^2\) sends \(v\) to \(w\). Thus an invariant space containing \(v\) contains all of \(v^\perp\). The action is faithful, proving effectiveness too. By AE.1 it is a group-type pair.

For \((2,2)\), AC.1 identifies \(\mathfrak{so}(4)\) as a sum of two compact simple rank-one ideals. Their exchange would have a three-dimensional fixed algebra by (AE.1), whereas the actual fixed algebra is \(\mathfrak{so}(2)\oplus\mathfrak{so}(2)\), of dimension two. Thus the involution preserves both ideals. It cannot be the identity on either three-dimensional ideal, since that ideal would already exceed the total fixed dimension. The rank-one classification AD.1 shows that it is the nonidentity rank-one involution on each. Hence this is exactly a product of two rank-one pairs. The excluded AII case \(n=1\) is the identity by AD.1. The excluded DIII case \(n=2\) has a four-dimensional fixed algebra in the two-factor \(\mathfrak{so}(4)\). It cannot exchange the factors, and AD.1 makes each factor's fixed dimension either one or three. The total four is therefore \(3+1\): a whole factor is fixed and commutes with the odd part in the other, so the pair is ineffective. It is not an additional irreducible transvection pair.

We now compute the centres of the fixed algebras. The algebras \(\mathfrak{su}(n)\), \(\mathfrak{sp}(n)\) and \(\mathfrak{so}(N)\), \(N\geq3\), have zero centre by AC.1. The boundary algebras \(\mathfrak{so}(1)\) and \(\mathfrak{so}(0)\) are zero, while \(\mathfrak{so}(2)\) is one-dimensional abelian. Commuting with the matrix units shows that \(Z(\mathfrak u(n))=i\mathbb RI_n\). In AIII the central scalar blocks satisfy the one real relation \(pa+qb=0\), so their centre is one dimensional, as proved in AD.1.

It follows that AI has one-dimensional isotropy centre only at \(n=2\), AII and CII have none, AIII, CI and DIII each have one, and an irreducible BDI entry has one precisely when a parameter is two. If both parameters are two, the centre has dimension two and the pair has just been shown reducible. The equivalence and uniqueness theorem E.2 now gives exactly the stated Hermitian entries and their two structures. This applies to either real dual by D.3. The group-type pairs of AE.1 have centre-free fixed algebra \(\mathfrak v\), so give no further nonflat irreducible Hermitian entries.

AD.1 already identifies AI at \(n=2\) with AIII at \(p=q=1\). AC.1 identifies \(C_1\) with \(A_1\), and the compact-form uniqueness AA.3 identifies their compact algebras. Their sole nonidentity involution class in AD.1 therefore identifies CI at \(n=1\) with the same rank-one pair. Its sphere/disc realization and scale were proved in L.4. D.1 proves that every invariant metric on an irreducible factor is a positive scalar multiple, and rescaling does not change the symmetry or the parallel complex structure. The table is an exhaustive algebraic list in these ranges; it does not assert that rows of different low-rank types never repeat, or replace the construction of their global homogeneous models. □

## AF. Exceptional toral involutions

We first treat the signed representatives of AB.2 whose diagram permutation is the identity. Write \(c_i=(-1)^{b_i}\), where \(b_i\) is either zero or one, and let \(\sigma_b\) denote this automorphism. AC.3 realizes it as a compact toral automorphism. Throughout this section the Cartan convention is
\[
a_{ij}=\alpha_j(h_i)=\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.
\tag{AF.1}
\]
The compact exceptional algebras of AA.1 are denoted \(\mathfrak g_2^c,\mathfrak f_4^c,\mathfrak e_6^c,\mathfrak e_7^c,\mathfrak e_8^c\). Their dimensions are respectively \(14,52,78,133,248\), by the root counts in [De Rham decomposition AE.4–AE.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-4) and the root-space theorem Q.3.

**Lemma AF.1 (the even-root fixed algebra).** For a root \(\alpha=\sum_i n_i\alpha_i\), put
\[
\chi_b(\alpha)=(-1)^{\sum_i b_i n_i},\qquad
R_b=\{\alpha\in R:\chi_b(\alpha)=1\}.
\tag{AF.2}
\]
Then, with \(\mathfrak h=\mathfrak t_{\mathbb C}\),
\[
(\mathfrak u^{\sigma_b})_{\mathbb C}
=\mathfrak h\oplus\bigoplus_{\alpha\in R_b}\mathfrak g_\alpha.
\tag{AF.3}
\]
Its compact centre has dimension \(r-\operatorname{rank}R_b\), its derived algebra has root system \(R_b\) in its span, and
\[
\dim_{\mathbb R}\mathfrak u^{\sigma_b}=r+|R_b|,
\qquad \dim_{\mathbb R}\mathfrak u_-=|R|-|R_b|.
\tag{AF.4}
\]

**Proof.** In the Serre presentation, \(\sigma_b\) fixes all \(h_i\), multiplies \(e_i\) by \((-1)^{b_i}\), and multiplies \(f_i\) by the same sign. A homogeneous bracket of root degree \(\sum_i n_i\alpha_i\) therefore has eigenvalue \((-1)^{\sum_i b_i n_i}\), including negative degrees. The homogeneous generation and exact root spaces proved in T.3–U.1 give this eigenvalue on every root space. They also show that every zero-weight vector belongs to \(\mathfrak h\). This proves (AF.3), and Q.3 proves the dimensions because every root space has complex dimension one.

The set \(R_b\) is reduced and closed under its own root reflections. Indeed if \(\alpha,\beta\in R_b\), then
\[
\chi_b(s_\alpha\beta)
=\chi_b(\beta)\chi_b(\alpha)^{-\beta(h_\alpha)}=1,
\]
and the exponent is an integer by Q.4. Thus the inherited inner product and root axioms make it a root system in its span.

AB.1 supplies the compact reductive splitting \(\mathfrak k=Z(\mathfrak k)\oplus\mathfrak s\), where \(\mathfrak k=\mathfrak u^{\sigma_b}\) and \(\mathfrak s=[\mathfrak k,\mathfrak k]\). The centralizer of \(\mathfrak h\) in (AF.3) is \(\mathfrak h\), so a central element must lie there. It is central precisely when every \(\alpha\in R_b\) vanishes on it. Consequently
\[
Z(\mathfrak k)_{\mathbb C}
=\{H\in\mathfrak h:\alpha(H)=0\text{ for all }\alpha\in R_b\}.
\tag{AF.5}
\]
Its dimension is \(r-\operatorname{rank}R_b\). Each root space in (AF.3) belongs to the derived algebra: choose \(H\in\mathfrak h\) with \(\alpha(H)\ne0\), and use \(X=\alpha(H)^{-1}[H,X]\). The coroot \(h_\alpha=[e_\alpha,f_\alpha]\) also belongs to it. The real span of these coroots is positive definite for the Killing form on \(i\mathfrak t\), by Q.5, and has dimension \(\operatorname{rank}R_b\). It is complementary to (AF.5). Comparing dimensions in the reductive splitting now gives
\[
\mathfrak s_{\mathbb C}
=\operatorname{span}_{\mathbb C}\{h_\alpha:\alpha\in R_b\}
\oplus\bigoplus_{\alpha\in R_b}\mathfrak g_\alpha.
\tag{AF.6}
\]
The coroot span is toral and self-centralizing in this algebra, since roots in its span do not vanish on all their coroots. Q.2–Q.5 therefore identify \(R_b\) as its root system. Its Euclidean metric may be rescaled separately on simple components without altering its Cartan matrix. U.1 identifies its complex simple factors, and AA.3 identifies their compact real forms. This also covers \(R_b=\varnothing\), when the derived algebra is zero. □

**Lemma AF.2 (ten exceptional fixed algebras).** Number the roots as follows. For \(G_2\), use the long-first basis
\[
\alpha_1=-2e_1+e_2+e_3,\qquad \alpha_2=e_1-e_2.
\tag{AF.7}
\]
For \(F_4\), use
\[
\alpha_1=e_2-e_3,\quad \alpha_2=e_3-e_4,\quad
\alpha_3=e_4,\quad
\alpha_4=(e_1-e_2-e_3-e_4)/2.
\tag{AF.8}
\]
For \(E_r\), \(r=6,7,8\), use the first \(r\) vectors of
\[
\begin{aligned}
\alpha_1&=(e_1+e_8-e_2-e_3-e_4-e_5-e_6-e_7)/2,\\
\alpha_2&=e_1+e_2,\qquad
\alpha_j=e_{j-1}-e_{j-2}\quad(3\leq j\leq8).
\end{aligned}
\tag{AF.9}
\]
The edges are \(1\!-\!3\!-\!4\!-\!5\!-\!6\!-\!7\!-\!8\) and \(2\!-\!4\), truncated at \(r\). If the single negative sign is at the vertex \(i\) in the following table, the fixed algebra and odd dimension are as displayed. Here \(\mathfrak{sp}(n)\) has real dimension \(n(2n+1)\).

| Type | \(i\) | Fixed compact algebra \(\mathfrak k\) | \(\dim\mathfrak k\) | \(\dim\mathfrak p\) |
| --- | --- | --- | ---: | ---: |
| \(G_2\) | 1 | \(\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)\) | 6 | 8 |
| \(F_4\) | 1 | \(\mathfrak{sp}(3)\oplus\mathfrak{sp}(1)\) | 24 | 28 |
| \(F_4\) | 4 | \(\mathfrak{so}(9)\) | 36 | 16 |
| \(E_6\) | 1 | \(\mathfrak{so}(10)\oplus\mathbb R\) | 46 | 32 |
| \(E_6\) | 2 | \(\mathfrak{su}(6)\oplus\mathfrak{sp}(1)\) | 38 | 40 |
| \(E_7\) | 1 | \(\mathfrak{so}(12)\oplus\mathfrak{sp}(1)\) | 69 | 64 |
| \(E_7\) | 2 | \(\mathfrak{su}(8)\) | 63 | 70 |
| \(E_7\) | 7 | \(\mathfrak e_6^c\oplus\mathbb R\) | 79 | 54 |
| \(E_8\) | 1 | \(\mathfrak{so}(16)\) | 120 | 128 |
| \(E_8\) | 8 | \(\mathfrak e_7^c\oplus\mathfrak{sp}(1)\) | 136 | 112 |

**Proof.** The explicit root systems and these bases were proved in [De Rham decomposition AE.4–AE.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-4). For \(G_2\), (AF.7) reverses the short-first basis there. Its six positive roots have coefficient pairs
\[
(1,0),\ (0,1),\ (1,1),\ (1,2),\ (1,3),\ (2,3).
\tag{AF.10}
\]
Even first coefficient leaves precisely \(\pm\alpha_2\) and \(\pm(2\alpha_1+3\alpha_2)\). Their inner product is zero, so the subsystem is \(A_1\oplus A_1\).

For \(F_4\), the roots are the 24 vectors \(\pm e_j\pm e_k\), the eight vectors \(\pm e_j\), and the sixteen half-vectors \((\pm e_1\pm e_2\pm e_3\pm e_4)/2\). The coefficient equations for (AF.8) include
\[
n_1=x_1+x_2,\qquad n_4=2x_1.
\tag{AF.11}
\]
When \(i=4\), all integer roots and no half-roots are even. This is exactly the \(B_4\) root system \(\{\pm e_j\pm e_k,\pm e_j\}\), with 32 roots.

When \(i=1\), an integer pair-root is even precisely when both indices are in \(\{1,2\}\) or both are in \(\{3,4\}\), giving eight roots. There are four even axis roots, \(\pm e_3,\pm e_4\). A half-root is even precisely when its first two signs are opposite, giving eight. Thus there are 20 even roots. The three roots \(\alpha_4,\alpha_3,\alpha_2\), in that order, have row Cartan matrix
\[
\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix},
\tag{AF.12}
\]
of type \(C_3\). The even root
\[
\theta_F=2\alpha_1+3\alpha_2+4\alpha_3+2\alpha_4=e_1+e_2
\tag{AF.13}
\]
is perpendicular to all three. Reflections in these four roots therefore generate a \(C_3\oplus A_1\) root subsystem with \(18+2=20\) roots. Here the count and identification follow from P.1 and [De Rham decomposition AE.3](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-3): the Gram matrix identifies each initial basis isometrically, up to its component scale, with the already proved coordinate model, and transports every generated reflection. All generated roots remain even by AF.1. Since there are exactly 20 even roots, this subsystem is all of \(R_b\).

For the \(E\)-systems, the common ambient description is
\[
R(E_8)=\{\pm e_j\pm e_k:j<k\}
\ \cup\
\left\{\tfrac12(\varepsilon_1,\ldots,\varepsilon_8):
\varepsilon_j=\pm1,\ \prod_j\varepsilon_j=1\right\}.
\tag{AF.14}
\]
The \(E_7\) roots satisfy \(x_7=-x_8\); the \(E_6\) roots additionally satisfy \(x_6=x_7\). Inverting (AF.9) gives
\[
\begin{aligned}
n_1&=2x_8,& n_8&=x_7+x_8,\\
n_7&=x_6+x_7+2x_8,&
n_4&=x_3+x_4+x_5+x_6+x_7+5x_8,\\
n_2&=(x_1+x_2+n_4)/2.
\end{aligned}
\tag{AF.15}
\]
These formulas hold in the smaller systems after imposing their coordinate equations and setting the absent coefficients to zero.

For \(i=1\), integer roots are even and half-roots are odd. In \(E_6\), the integer roots are exactly \(\pm e_j\pm e_k\) with \(1\leq j<k\leq5\), a \(D_5\) system with 40 roots and rank five. In \(E_7\), they are the \(D_6\) roots on the first six coordinates together with \(\pm(e_7-e_8)\), perpendicular to them. This is \(D_6\oplus A_1\), with \(60+2=62\) roots and rank seven. In \(E_8\), they form \(D_8\), with 112 roots and rank eight.

For \(E_6\) at \(i=2\), (AF.15) becomes
\[
n_2=(x_1+x_2+x_3+x_4+x_5+3x_8)/2.
\tag{AF.16}
\]
Among the integer roots on the first five coordinates, exactly the opposite-sign pairs are even, giving \(5\cdot4=20\) roots. For a half-root, let \(m\) be the number of negative signs among its first five coordinates, and put \(x_8=t/2\). The product condition in (AF.14) gives \(t=(-1)^m\); its last three signs are \((-t,-t,t)\). Thus
\[
n_2=\frac{5-2m+3(-1)^m}{4}.
\tag{AF.17}
\]
This is even for \(m=0,1,4,5\), and odd for \(m=2,3\). The even half-roots therefore number
\(\binom50+\binom51+\binom54+\binom55=12\). There are \(20+12=32\) even roots. The roots
\[
\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6
\]
form an \(A_5\) chain, and
\[
\theta_6=\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6
\tag{AF.18}
\]
is an even root perpendicular to that chain. To check both assertions directly, substitute (AF.9): \(\theta_6=(1,1,1,1,1,-1,-1,1)/2\). Its scalar products with the five chain vectors are zero, and its squared length is two. Thus their generated subsystem is \(A_5\oplus A_1\), with \(30+2=32\) roots. The same reflection and count argument as for (AF.12) proves equality with all even roots.

For \(E_7\) at \(i=2\),
\[
n_2=(x_1+\cdots+x_6+4x_8)/2.
\tag{AF.19}
\]
The opposite-sign pair-roots on the first six coordinates give 30 even integer roots, and both roots \(\pm(e_7-e_8)\) are even too. For a half-root, write \(x_8=t/2\) and let \(m\) count negatives among the first six signs. Their product is \(-1\), so \(m=1,3,5\), and
\[
n_2=(3-m)/2+t.
\tag{AF.20}
\]
It is even precisely for \(m=1,5\). There are \(2(\binom61+\binom65)=24\) even half-roots, giving 56 even roots in total. The seven even roots
\[
\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7,\beta_7,
\quad
\beta_7=\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6
\tag{AF.21}
\]
form an \(A_7\) chain in the displayed order. Indeed \(\beta_7=(1,1,1,1,1,-1,-1,1)/2\) has scalar product \(-1\) with \(\alpha_7=e_6-e_5\) and zero with the first five displayed roots; the other edges follow from (AF.9). Their Gram matrix is positive definite, so they are independent. They generate 56 roots by the proved \(A_7\) coordinate model. All are even, and the count again proves equality.

For \(E_7\) at \(i=7\), its ambient constraint makes \(n_7=x_6+x_8\). An even integer root has both indices among the first five, giving 40. A half-root is even exactly when \(\varepsilon_6=-\varepsilon_8\), giving 32. More intrinsically, the even roots are exactly those satisfying \(x_6=x_7=-x_8\): the \(E_6\) system in (AF.14), of rank six and with 72 roots.

Finally, for \(E_8\) at \(i=8\), evenness is the condition \(x_7+x_8\in2\mathbb Z\). Its integer roots are the 60 pair-roots on the first six coordinates and the four roots \(\pm e_7\pm e_8\). Its even half-roots have \(\varepsilon_7=-\varepsilon_8\), so there are 64. These 128 roots are precisely the \(E_7\) system \(x_7=-x_8\), together with \(\pm(e_7+e_8)\), perpendicular to it. Thus the subsystem is \(E_7\oplus A_1\).

We have exhibited the entire subsystem in each case, not only a subalgebra of the indicated dimension. AF.1 now gives the derived compact algebra by U.1 and AA.3. The rank is full except in the \(D_5\subset E_6\) and \(E_6\subset E_7\) cases, where its deficit is one; these give exactly the two \(\mathbb R\) centres. The classical names follow from AC.1, including \(A_1=C_1=\mathfrak{sp}(1)_{\mathbb C}\). Adding rank to the even-root counts gives every fixed dimension in the table, and subtracting it from \(14,52,78,133,248\) gives every odd dimension. □

**Lemma AF.3 (exhaustion of the exceptional toral signs).** Up to compact-automorphism conjugacy, every sign choice with identity diagram permutation is the all-positive choice or one of the ten single-negative choices of AF.2. Within each exceptional type, no two choices in this list are conjugate.

**Proof.** AC.3 proves that conjugation by an actual compact reflection changes the bits by
\[
b'_j=b_j+(a_{ij}\bmod2)b_i\pmod2.
\tag{AF.22}
\]
Thus a move at a white vertex changes nothing; a move at a black vertex toggles exactly the odd-entry neighbours in row \(i\). We give the complete finite reduction, including a predecessor for every nonzero colouring.

Encode a colouring by \(m=\sum_{i=1}^r2^{i-1}b_i\). In the tables below write \(m\) in two hexadecimal digits: \(a,b,c,d,e,f\) mean \(10,11,12,13,14,15\). Leading zeroes have no effect. Let \(N_i\) be the mask of the odd-entry neighbours. The move is
\[
T_i(m)=
\begin{cases}
m\mathbin{\mathrm{xor}}N_i,&\text{if bit }i\text{ of }m\text{ is }1,\\
m,&\text{otherwise}.
\end{cases}
\tag{AF.23}
\]
Here xor means addition in each binary digit modulo two. The masks, in increasing vertex order, are:

| Type | \(N_1,\ldots,N_r\), in hexadecimal |
| --- | --- |
| \(G_2\) | 02, 01 |
| \(F_4\) | 02, 05, 08, 04 |
| \(E_6\) | 04, 08, 09, 16, 28, 10 |
| \(E_7\) | 04, 08, 09, 16, 28, 50, 20 |
| \(E_8\) | 04, 08, 09, 16, 28, 50, a0, 40 |

These follow directly from (AF.7)–(AF.9) and (AF.1). In particular the \(F_4\) move at vertex 3 does not toggle vertex 2, since \(a_{32}=-2\); the reverse entry \(a_{23}=-1\) is odd.

Each block below starts with its representative in row \(d=0\). An entry \(m/i\) in row \(d>0\) states that bit \(i\) of \(m\) is one and \(T_i(m)\) occurs in row \(d-1\) of that same block. For example, \(05/1\) in the \(E_6\) block goes to \(05\mathbin{\mathrm{xor}}04=01\). Each entry can therefore be checked by one xor and a lookup in the immediately preceding row.

The G2 block with representative 01 (3 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 01 |
| 1 | 03/1 |
| 2 | 02/2 |

The F4 block with representative 01 (12 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 01 |
| 1 | 03/1 |
| 2 | 06/2 |
| 3 | 0e/3 |
| 4 | 0a/4, 0b/2 |
| 5 | 09/1, 0f/2 |
| 6 | 07/3, 0d/1 |
| 7 | 02/2, 05/1 |

The F4 block with representative 08 (3 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 08 |
| 1 | 0c/4 |
| 2 | 04/3 |

The E6 block with representative 01 (27 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 01 |
| 1 | 05/1 |
| 2 | 0c/3 |
| 3 | 1a/4 |
| 4 | 12/2, 32/5 |
| 5 | 22/6, 3a/2 |
| 6 | 2a/2, 2c/4 |
| 7 | 25/3, 3c/4 |
| 8 | 14/5, 21/1, 35/3 |
| 9 | 1d/3, 31/1 |
| 10 | 0b/4, 19/1 |
| 11 | 03/2, 0f/1 |
| 12 | 06/3, 07/1 |
| 13 | 0e/2 |
| 14 | 18/4 |
| 15 | 30/5 |
| 16 | 20/6 |

The E6 block with representative 02 (36 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 02 |
| 1 | 0a/2 |
| 2 | 1c/4 |
| 3 | 15/3, 34/5 |
| 4 | 11/1, 24/6, 3d/3 |
| 5 | 2b/4, 2d/3, 39/1 |
| 6 | 23/2, 29/1, 2f/1, 3b/4 |
| 7 | 13/5, 26/3, 27/1, 33/2, 3f/1 |
| 8 | 17/1, 1b/2, 2e/2, 36/3, 37/1 |
| 9 | 0d/4, 1e/3, 1f/1, 38/4, 3e/2 |
| 10 | 04/3, 08/4, 09/1, 10/5, 16/2, 28/4 |

The E7 block with representative 01 (63 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 01 |
| 1 | 05/1 |
| 2 | 0c/3 |
| 3 | 1a/4 |
| 4 | 12/2, 32/5 |
| 5 | 3a/2, 62/6 |
| 6 | 2c/4, 42/7, 6a/2 |
| 7 | 25/3, 4a/2, 7c/4 |
| 8 | 21/1, 54/5, 5c/4, 75/3 |
| 9 | 55/3, 5d/3, 71/1, 74/5 |
| 10 | 24/6, 4b/4, 51/1, 59/1, 7d/3 |
| 11 | 2d/3, 43/2, 4f/1, 6b/4, 79/1 |
| 12 | 29/1, 3b/4, 46/3, 47/1, 63/2, 6f/1 |
| 13 | 13/5, 33/2, 3f/1, 4e/2, 66/3, 67/1 |
| 14 | 17/1, 1b/2, 36/3, 37/1, 58/4, 6e/2 |
| 15 | 0d/4, 1e/3, 1f/1, 3e/2, 70/5, 78/4 |
| 16 | 04/3, 08/4, 09/1, 16/2, 20/6, 28/4, 50/5 |

The E7 block with representative 02 (36 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 02 |
| 1 | 0a/2 |
| 2 | 1c/4 |
| 3 | 15/3, 34/5 |
| 4 | 11/1, 3d/3, 64/6 |
| 5 | 2b/4, 39/1, 44/7, 6d/3 |
| 6 | 23/2, 2f/1, 4d/3, 69/1, 7b/4 |
| 7 | 26/3, 27/1, 49/1, 53/5, 5b/4, 73/2, 7f/1 |
| 8 | 2e/2, 57/1, 5f/1, 76/3, 77/1 |
| 9 | 38/4, 56/3, 5e/3, 7e/2 |
| 10 | 10/5, 48/4, 68/4 |

The E7 block with representative 40 (28 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 40 |
| 1 | 60/7 |
| 2 | 30/6 |
| 3 | 18/5 |
| 4 | 0e/4 |
| 5 | 06/2, 07/3 |
| 6 | 03/1, 0f/2 |
| 7 | 0b/1, 19/4 |
| 8 | 1d/1, 31/5 |
| 9 | 14/3, 35/1, 61/6 |
| 10 | 3c/3, 41/7, 65/1 |
| 11 | 2a/4, 45/1, 6c/3 |
| 12 | 22/2, 4c/3, 7a/4 |
| 13 | 52/5, 5a/4, 72/2 |

The E8 block with representative 01 (135 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 01 |
| 1 | 05/1 |
| 2 | 0c/3 |
| 3 | 1a/4 |
| 4 | 12/2, 32/5 |
| 5 | 3a/2, 62/6 |
| 6 | 2c/4, 6a/2, c2/7 |
| 7 | 25/3, 7c/4, 82/8, ca/2 |
| 8 | 21/1, 54/5, 75/3, 8a/2, dc/4 |
| 9 | 5d/3, 71/1, 9c/4, d5/3, f4/5 |
| 10 | 4b/4, 59/1, 95/3, a4/6, b4/5, d1/1, fd/3 |
| 11 | 43/2, 4f/1, 91/1, ad/3, bd/3, e4/6, eb/4, f9/1 |
| 12 | 44/7, 46/3, 47/1, a9/1, ab/4, b9/1, bb/4, e3/2, ed/3, ef/1 |
| 13 | 4d/3, 4e/2, 93/5, a3/2, af/1, b3/2, bf/1, e6/3, e7/1, e9/1, fb/4 |
| 14 | 49/1, 58/4, 5b/4, 97/1, 9b/2, a6/3, a7/1, b6/3, b7/1, d3/5, ee/2, f3/2, ff/1 |
| 15 | 53/2, 5f/1, 70/5, 73/5, 8d/4, 9e/3, 9f/1, ae/2, be/2, d7/1, db/2, f6/3, f7/1, f8/4 |
| 16 | 20/6, 23/6, 56/3, 57/1, 77/1, 7b/2, 84/3, 88/4, 89/1, 96/2, a8/4, b8/4, cd/4, d0/5, de/3, df/1, fe/2 |
| 17 | 27/1, 2b/2, 5e/2, 6d/4, 7e/3, 7f/1, 90/5, c4/3, c8/4, c9/1, d6/2, e8/4 |
| 18 | 2e/3, 2f/1, 3d/4, 48/4, 64/3, 68/4, 69/1, 76/2 |
| 19 | 15/5, 26/2, 34/3, 38/4, 39/1 |
| 20 | 10/5, 11/1, 1c/3 |
| 21 | 0a/4 |
| 22 | 02/2 |

The E8 block with representative 80 (120 colourings):

| \(d\) | Colouring / reflection |
| ---: | --- |
| 0 | 80 |
| 1 | c0/8 |
| 2 | 60/7 |
| 3 | 30/6 |
| 4 | 18/5 |
| 5 | 0e/4 |
| 6 | 06/2, 07/3 |
| 7 | 03/1, 0f/2 |
| 8 | 0b/1, 19/4 |
| 9 | 1d/1, 31/5 |
| 10 | 14/3, 35/1, 61/6 |
| 11 | 3c/3, 65/1, c1/7 |
| 12 | 2a/4, 6c/3, 81/8, c5/1 |
| 13 | 22/2, 7a/4, 85/1, cc/3 |
| 14 | 52/5, 72/2, 8c/3, da/4 |
| 15 | 5a/2, 9a/4, d2/2, f2/5 |
| 16 | 4c/4, 92/2, a2/6, b2/5, fa/2 |
| 17 | 45/3, aa/2, ba/2, e2/6, ec/4 |
| 18 | 41/1, 42/7, ac/4, bc/4, e5/3, ea/2 |
| 19 | 4a/2, 94/5, a5/3, b5/3, e1/1, fc/4 |
| 20 | 5c/4, 9d/3, a1/1, b1/1, d4/5, f5/3 |
| 21 | 55/3, 74/5, 8b/4, 99/1, dd/3, f1/1 |
| 22 | 24/6, 51/1, 7d/3, 83/2, 8f/1, cb/4, d9/1 |
| 23 | 2d/3, 6b/4, 79/1, 86/3, 87/1, c3/2, cf/1 |
| 24 | 29/1, 3b/4, 63/2, 6f/1, 8e/2, c6/3, c7/1 |
| 25 | 13/5, 33/2, 3f/1, 66/3, 67/1, 98/4, ce/2 |
| 26 | 17/1, 1b/2, 36/3, 37/1, 6e/2, b0/5, d8/4 |
| 27 | 0d/4, 1e/3, 1f/1, 3e/2, 78/4, e0/6, f0/5 |
| 28 | 04/3, 08/4, 09/1, 16/2, 28/4, 40/7, 50/5, a0/6 |


For completeness, these finite certificates require only two kinds of integer checks. First, in each type the masks listed in all its blocks are distinct and, in increasing order, are exactly \(01,\ldots,2^r-1\). The block sizes are respectively
\[
3;\qquad 12+3;\qquad27+36;\qquad63+36+28;\qquad135+120.
\tag{AF.24}
\]
Second, every displayed reflection has its bit set and its xor target in the preceding row, by the neighbour-mask table and (AF.23). The explicit masks and reflection indices above give all these checks without a root-classification assumption or an omitted search. Induction on \(d\) now conjugates every entry to its block representative: apply its displayed reflection and then the already established chain for the preceding row. Each move is conjugation by AC.3. The zero colouring gives the identity. This proves exhaustion.

The fixed-algebra dimensions in AF.2 are different within each type, and all are smaller than the dimension of the whole algebra, which is fixed by the identity. Conjugate linear maps have isomorphic fixed algebras and hence equal fixed dimensions. Therefore none of the listed representatives can be conjugate to another. This last argument also excludes a merger by compact automorphisms outside the subgroup generated by these particular reflections; no assertion that a normalizer equals a Weyl group is needed. □

## AG. The exceptional pairs with toral involution

For a compact involution, write \(\mathfrak u=\mathfrak k\oplus\mathfrak p\) and \(\mathfrak g_{\mathbb R}=\mathfrak k\oplus i\mathfrak p\) for its real dual. In this section a subscript on an exceptional real form records the signature difference of its Killing form: the number of positive directions minus the number of negative directions. It is not a dimension.

**Theorem AG.1 (the ten toral exceptional pairs).** The nonidentity exceptional toral involutions give the following compact fixed algebras and real duals. Every row is effective and irreducible.

| Family | \(\mathfrak u\) | \(\mathfrak k\) | Real dual | \(\dim\mathfrak p\) |
| --- | --- | --- | --- | ---: |
| G | \(\mathfrak g_2^c\) | \(\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)\) | \(\mathfrak g_{2(2)}\) | 8 |
| FI | \(\mathfrak f_4^c\) | \(\mathfrak{sp}(3)\oplus\mathfrak{sp}(1)\) | \(\mathfrak f_{4(4)}\) | 28 |
| FII | \(\mathfrak f_4^c\) | \(\mathfrak{so}(9)\) | \(\mathfrak f_{4(-20)}\) | 16 |
| EII | \(\mathfrak e_6^c\) | \(\mathfrak{su}(6)\oplus\mathfrak{sp}(1)\) | \(\mathfrak e_{6(2)}\) | 40 |
| EIII | \(\mathfrak e_6^c\) | \(\mathfrak{so}(10)\oplus\mathbb R\) | \(\mathfrak e_{6(-14)}\) | 32 |
| EV | \(\mathfrak e_7^c\) | \(\mathfrak{su}(8)\) | \(\mathfrak e_{7(7)}\) | 70 |
| EVI | \(\mathfrak e_7^c\) | \(\mathfrak{so}(12)\oplus\mathfrak{sp}(1)\) | \(\mathfrak e_{7(-5)}\) | 64 |
| EVII | \(\mathfrak e_7^c\) | \(\mathfrak e_6^c\oplus\mathbb R\) | \(\mathfrak e_{7(-25)}\) | 54 |
| EVIII | \(\mathfrak e_8^c\) | \(\mathfrak{so}(16)\) | \(\mathfrak e_{8(8)}\) | 128 |
| EIX | \(\mathfrak e_8^c\) | \(\mathfrak e_7^c\oplus\mathfrak{sp}(1)\) | \(\mathfrak e_{8(-24)}\) | 112 |

Together with the identity, these exhaust all compact involutions for \(G_2,F_4,E_7,E_8\). For \(E_6\) they exhaust the representatives with identity diagram permutation. Among these ten rows, precisely EIII and EVII are Hermitian, of complex dimensions 16 and 27 respectively. Each has the two invariant complex structures differing by sign.

**Proof.** A toral involution fixes a maximal abelian subalgebra pointwise. With its adapted generators as in AB.2, its eigenvalues on the simple-root spaces are signs and its diagram permutation is the identity. Conversely every such sign choice is realized by AC.3, so AF.3 exhausts these involutions and distinguishes their classes. AF.2 supplies every fixed algebra and dimension in the table.

For \(G_2\) and \(F_4\), an automorphism of the Cartan diagram preserves root lengths. In the two-vertex \(G_2\) diagram the lengths distinguish the vertices. In the four-vertex \(F_4\) chain, the only nonidentity graph automorphism reverses the chain and exchanges long and short roots, so it does not preserve the row Cartan matrix. The \(E_7\) and \(E_8\) diagrams each have a unique vertex of valence three. The three legs from it have lengths \(1,2,3\) and \(1,2,4\), respectively. A diagram automorphism fixes that vertex and preserves leg length, hence fixes every vertex. Therefore AB.2 has no nonidentity diagram permutation for these four types, and its existence theorem proves the asserted full exhaustion there. The \(E_6\) legs have lengths \(1,2,2\), so their exchange remains possible; the present assertion leaves those representatives outside its scope.

Every compact algebra in the table is simple by AE.1. Each displayed involution has positive odd dimension by AF.2 and is therefore not the identity. AE.1 proves effectiveness and irreducibility. D.3 constructs the dual bracket, while AA.5 proves that \(\mathfrak k\oplus i\mathfrak p\) is the corresponding real form. The Killing form is negative definite on \(\mathfrak k\), positive definite on \(i\mathfrak p\), and these subspaces are orthogonal, as proved in AA.5. Thus its signature difference is
\[
\dim\mathfrak p-\dim\mathfrak k.
\tag{AG.1}
\]
For the ten rows in order, the arithmetic is
\[
\begin{gathered}
8-6=2,\quad28-24=4,\quad16-36=-20,\\
40-38=2,\quad32-46=-14,\quad70-63=7,\\
64-69=-5,\quad54-79=-25,\quad128-120=8,\quad112-136=-24.
\end{gathered}
\tag{AG.2}
\]
These are the subscripts used in the table. Its real duals are thus defined and constructed by the displayed compact pairs; no separate classification is hidden in the names. AA.5 also shows that inequivalent compact involutions give nonisomorphic corresponding real forms.

Finally AF.1–AF.2 give a one-dimensional centre precisely for the \(D_5\oplus\mathbb R\) and \(E_6\oplus\mathbb R\) fixed algebras. All the others have zero centre. The necessary and sufficient criterion and uniqueness assertion E.2 therefore give exactly EIII and EVII among these rows, with two complex structures differing by sign on either real dual. Their real odd dimensions are 32 and 54, so their complex dimensions are 16 and 27. D.1 proves that every invariant metric is a positive scalar multiple on each irreducible pair. No global quotient identification is needed for, or asserted by, this algebraic classification. □

## AH. The two E6 diagram-reversal cases

Use the \(E_6\) numbering of (AF.9). Its nonidentity diagram permutation is
\[
\pi=(1\ 6)(3\ 5),\qquad \pi(2)=2,\quad\pi(4)=4.
\tag{AH.1}
\]
Let \(\Gamma\) be the compact automorphism which applies \(\pi\) to each of the normalized Serre generators \(h_i,e_i,f_i\). It exists and has square one by AB.2. We write \(\mathfrak h_+\) for its fixed Cartan subspace, of complex dimension four.

**Lemma AH.1 (the fixed algebra of pure diagram reversal).** The fixed compact algebra \((\mathfrak e_6^c)^\Gamma\) is \(\mathfrak f_4^c\). All 24 roots fixed by the diagram permutation have \(\Gamma=+1\) on their root spaces. The other 48 roots form 24 two-element orbits, each contributing one root space to the fixed algebra.

**Proof.** We first compute the restrictions of every root to \(\mathfrak h_+\), including multiplicities; this will also supply an upper bound for the fixed dimension. Use the Euclidean root space and the coordinate \(E_6\) model of [De Rham decomposition AE.4](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-4). The permutation \(\pi\) acts orthogonally because it preserves the simple-root Gram matrix. Its fixed root subspace is the image of the orthogonal projection \(P=(1+\pi)/2\). By the Killing duality in Q.5, restricting roots to \(\mathfrak h_+\) corresponds to this projection.

Put
\[
\beta_1=(\alpha_1+\alpha_6)/2,\quad
\beta_2=(\alpha_3+\alpha_5)/2,\quad
\beta_3=\alpha_4,\quad \beta_4=\alpha_2.
\tag{AH.2}
\]
Their squared lengths are \(1,1,2,2\). Their only nonzero off-diagonal scalar products are
\[
(\beta_1,\beta_2)=-\tfrac12,\qquad
(\beta_2,\beta_3)=-1,\qquad
(\beta_3,\beta_4)=-1.
\tag{AH.3}
\]
Thus the isometry sending them, in order, to
\[
\tfrac12(e_1-e_2-e_3-e_4),\quad e_4,\quad
e_3-e_4,\quad e_2-e_3
\tag{AH.4}
\]
identifies the projected space with \(\mathbb R^4\). Indeed the four vectors in (AH.4) are independent, and their Gram matrix is exactly (AH.3), so the indicated linear map preserves the inner product on a basis.

For \(\alpha=\sum_{i=1}^6n_i\alpha_i\), put
\[
u=n_1+n_6,\quad v=n_3+n_5,\quad w=n_4,\quad z=n_2.
\]
Its projected coordinates in (AH.4) are
\[
y=\left(\frac u2,\ z-\frac u2,\ w-z-\frac u2,\
v-w-\frac u2\right).
\tag{AH.5}
\]
In the original eight-dimensional coordinates \(x\), the \(E_6\) constraint is \(x_6=x_7=-x_8\). Solving (AF.9), or substituting its simple vectors in both sides of (AH.5), gives
\[
\begin{aligned}
y_1&=(x_5+3x_8)/2,\\
y_2&=(x_1+x_2+x_3+x_4)/2,\\
y_3&=(-x_1-x_2+x_3+x_4)/2,\\
y_4&=(-x_1+x_2-x_3+x_4)/2.
\end{aligned}
\tag{AH.6}
\]

The 72 roots are the 40 integer vectors \(\pm e_i\pm e_j\), \(1\leq i<j\leq5\), and the 32 half-vectors
\[
x=\tfrac12(\varepsilon_1,\ldots,\varepsilon_5,-t,-t,t),
\qquad \varepsilon_i=\pm1,\quad
t=\prod_{i=1}^5\varepsilon_i,
\tag{AH.7}
\]
as proved in that earlier coordinate lesson. We enumerate their projections without assuming a folding theorem. In the last three coordinates of \(y\), the images of \(e_1,e_2,e_3,e_4\) are respectively
\[
\tfrac12(1,-1,-1),\quad
\tfrac12(1,-1,1),\quad
\tfrac12(1,1,-1),\quad
\tfrac12(1,1,1).
\tag{AH.8}
\]
For the 24 integer roots whose indices are both at most four, take the sums and differences of the six distinct pairs in (AH.8), and their negatives. Their images are the twelve vectors \(\pm e_a\pm e_b\), \(2\leq a<b\leq4\), each once, and the six vectors \(\pm e_a\), \(2\leq a\leq4\), each twice. This pair calculation can be read directly from (AH.8): the sums, before their negatives, are
\[
(1,-1,0),\ (1,0,-1),\ (1,0,0),\
(1,0,0),\ (1,0,1),\ (1,1,0),
\]
and the differences in the same pair order \(12,13,14,23,24,34\) are
\[
(0,0,-1),\ (0,-1,0),\ (0,-1,-1),\
(0,-1,1),\ (0,-1,0),\ (0,0,-1).
\]
Their first projected coordinate is zero. The remaining sixteen integer roots have one index equal to five. That coordinate gives \(y_1=\pm1/2\), independently of the signs of the other index. The four vectors (AH.8) and their negatives give all eight sign triples with entries \(\pm1/2\). Thus these sixteen roots give all sixteen half-vectors in \(\mathbb R^4\), each once.

For (AH.7), the first projected coordinate is \((\varepsilon_5+3t)/4\). If \(\varepsilon_5=t\), then \(\prod_{i=1}^4\varepsilon_i=1\) and \(y_1=t\). The eight even-sign patterns on the first four coordinates give, in the remaining three coordinates, the six axis vectors \(\pm e_2,\pm e_3,\pm e_4\), each once, and zero twice. Explicitly, the patterns \(++++,++--,+-+-\) and their negatives give the three opposite axis pairs; the patterns \(+--+\) and \(-++-\) give zero by (AH.6). Varying \(t\) therefore produces the twelve vectors \(\pm e_1\pm e_a\), \(a=2,3,4\), each once, and \(\pm e_1\), each twice.

If \(\varepsilon_5=-t\), the first four signs have product \(-1\) and \(y_1=t/2\). The four patterns with exactly one minus sign give, in the remaining three coordinates,
\[
\tfrac12(1,1,1),\quad\tfrac12(1,1,-1),\quad
\tfrac12(1,-1,1),\quad\tfrac12(1,-1,-1),
\]
in order of the position of that minus sign; the patterns with three minus signs give their negatives. These are all eight half-sign triples. Varying \(t\) produces all sixteen half-vectors a second time.

We have therefore proved that the nonzero projected weights are exactly
\[
\begin{aligned}
L&=\{\pm e_i\pm e_j:1\leq i<j\leq4\},\\
S&=\{\pm e_i:1\leq i\leq4\}
\cup\{\tfrac12(\pm e_1\pm e_2\pm e_3\pm e_4)\}.
\end{aligned}
\tag{AH.9}
\]
There are 24 weights of each kind. Every long weight has multiplicity one, and every short weight has multiplicity two. No root projects to zero. A root has norm squared two, so equality of its projected norm with two holds exactly when it is fixed by the orthogonal projection. Consequently the 24 roots over \(L\) are precisely the \(\pi\)-fixed roots. Every weight in \(S\) comes from two distinct roots interchanged by \(\pi\), because projection is \(\pi\)-invariant and its multiplicity is exactly two.

The action of \(\Gamma\) interchanges the two one-dimensional spaces of each such pair and has square one. Its fixed space on that two-dimensional sum has dimension one: if \(X\) is in one root space, \(X+\Gamma X\) spans it. On each diagram-fixed root space it acts by a sign, not yet determined. On the Cartan its fixed dimension is four. It follows already that
\[
\dim_{\mathbb C}\mathfrak g^\Gamma\leq4+24+24=52.
\tag{AH.10}
\]

We now construct a subalgebra attaining this bound. For the four orbits
\[
O_1=\{1,6\},\quad O_2=\{3,5\},\quad
O_3=\{4\},\quad O_4=\{2\},
\]
define
\[
H_a=\sum_{i\in O_a}h_i,\qquad
E_a=\sum_{i\in O_a}e_i,\qquad
F_a=\sum_{i\in O_a}f_i.
\tag{AH.11}
\]
Every vector is fixed by \(\Gamma\). The Serre relations in \(\mathfrak e_{6,\mathbb C}\) give
\[
\begin{aligned}
[H_a,H_b]&=0,& [E_a,F_b]&=\delta_{ab}H_a,\\
[H_a,E_b]&=B_{ab}E_b,&[H_a,F_b]&=-B_{ab}F_b.
\end{aligned}
\tag{AH.12}
\]
where the row Cartan matrix is
\[
B=\begin{pmatrix}
2&-1&0&0\\
-1&2&-2&0\\
0&-1&2&-1\\
0&0&-1&2
\end{pmatrix}.
\tag{AH.13}
\]
For example, the two vertices in \(O_2\) each neighbour vertex 4, giving \(B_{23}=-2\); vertex 4 pairs with either member of \(O_2\) with value \(-1\), giving \(B_{32}=-1\). The other entries follow from the displayed \(E_6\) edges. The four \(H_a\) are independent and span \(\mathfrak h_+\).

It remains to check the positive and negative Serre relations, not merely (AH.12). The restricted weight of \(E_a\) is \(\beta_a\). For \(a\ne b\), the weight of
\((\operatorname{ad}E_a)^{1-B_{ab}}E_b\) is
\((1-B_{ab})\beta_a+\beta_b\). Its norm squared is
\[
(1-B_{ab})(\beta_a,\beta_a)+(\beta_b,\beta_b).
\tag{AH.14}
\]
This follows by expanding the square and using
\(2(\beta_a,\beta_b)=B_{ab}(\beta_a,\beta_a)\).
The value is at least three: the two short roots are adjacent, so a zero entry never joins two short roots, and every negative entry makes the indicated sum at least three. But (AH.9) lists all possible nonzero restricted weights in the ambient algebra, and all have norm squared one or two. The zero-weight space is \(\mathfrak h\); it cannot contain a vector of the displayed nonzero weight. Therefore every positive Serre bracket vanishes. The same argument for the negative weight proves the negative Serre relations.

Matrix (AH.13) is the \(F_4\) matrix for the reversed basis (AH.4), proved in [De Rham decomposition AE.5](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-5). U.1 now gives a homomorphism from the complex simple \(F_4\) algebra into \(\mathfrak g^\Gamma\), mapping its generators to (AH.11). It is nonzero because \(H_1\ne0\). Its kernel is an ideal of a simple algebra, hence zero. That algebra has dimension \(4+48=52\), by its complete root count. The upper bound (AH.10) proves that this injection is onto.

Equality also forces every one of the 24 diagram-fixed root spaces to contribute one fixed dimension, so its \(\Gamma\)-sign is \(+1\). On the generators, compact conjugation satisfies
\(\tau H_a=-H_a,\ \tau E_a=-F_a,\ \tau F_a=-E_a\). The presentation isomorphism therefore intertwines it with the compact conjugation of AA.1. Taking real fixed points gives precisely the compact form \(\mathfrak f_4^c\), proving all assertions. □

**Lemma AH.2 (the signed diagram reversal has fixed algebra \(C_4\)).** Let \(d\) be the toral automorphism with \(d(e_2)=-e_2\) and \(d(e_j)=e_j\) for \(j\ne2\), with the inverse signs on the \(f_j\). The compact involution
\(\sigma=d\Gamma\) has fixed algebra \(\mathfrak{sp}(4)\), of dimension 36.

**Proof.** The sign occurs at a vertex fixed by \(\pi\), so \(d\) commutes with \(\Gamma\); both have square one. Existence and compactness are AB.2 and AC.3. The action on the Cartan is still \(\pi\), giving the fixed toral space \(\mathfrak h_+\).

On a root of coefficients \(n_i\), \(d\) acts by \((-1)^{n_2}\), by AF.1. The two roots of a nonfixed \(\pi\)-orbit have the same \(n_2\). Thus on their two-dimensional sum, \(\sigma\) is a common sign times the exchange involution \(\Gamma\), and still has a one-dimensional fixed space. Its \(\mathfrak h_+\)-weight is unchanged. All 24 short weights \(S\) in (AH.9) therefore occur once in the fixed algebra.

On a diagram-fixed root, AH.1 proves that \(\Gamma\) is the identity. That root space survives precisely when \(n_2\) is even. Equation (AH.5) gives the exact identity
\[
n_2=y_1+y_2.
\tag{AH.15}
\]
For the long roots \(L=\{\pm e_i\pm e_j\}\), this sum is even precisely when both indices lie in \(\{1,2\}\) or both in \(\{3,4\}\). There are eight surviving long roots:
\[
L_0=\{\pm e_1\pm e_2,\ \pm e_3\pm e_4\}.
\tag{AH.16}
\]
Consequently the fixed algebra has Cartan part \(\mathfrak h_+\) of dimension four and exactly the 32 nonzero weights \(S\cup L_0\), each one dimensional. There are no further zero-weight vectors: AH.9 shows that no ambient root restricts to zero, and the fixed part of the ambient Cartan is exactly \(\mathfrak h_+\). In particular this toral subalgebra is self-centralizing.

To identify the complete root system, take the mutually orthogonal vectors
\[
\begin{aligned}
f_1&=(e_1+e_2)/2,& f_2&=(e_1-e_2)/2,\\
f_3&=(e_3+e_4)/2,& f_4&=(e_3-e_4)/2.
\end{aligned}
\tag{AH.17}
\]
Each has norm squared \(1/2\). The eight vectors in (AH.16) are exactly \(\pm2f_i\). The vectors \(\pm f_1\pm f_2,\ \pm f_3\pm f_4\) are the eight coordinate axes \(\pm e_i\); and the sixteen vectors \(\pm f_i\pm f_j\), with \(i\in\{1,2\}\) and \(j\in\{3,4\}\), are all sixteen half-vectors. Thus
\[
S\cup L_0=\{\pm2f_i,\ \pm f_i\pm f_j:i<j\}.
\tag{AH.18}
\]
This is exactly the \(C_4\) root system of [De Rham decomposition AE.3](holonomy-and-the-de-rham-decomposition-theorem.md#lemma-ae-3), scaled uniformly by \(1/\sqrt2\).

We justify that these weights identify the Lie algebra, including its centre. AB.1 makes the fixed compact algebra reductive. A central element in its complexification commutes with the self-centralizing Cartan \(\mathfrak h_+\), so belongs to that Cartan, and every weight in (AH.18) vanishes on it. These weights span its full dual, hence the central element is zero. The fixed algebra is therefore compact semisimple. Q.2–Q.5 identify its displayed Cartan weights as its roots, and U.1 identifies its complex algebra as type \(C_4\). Here the inner product inherited from the ambient Killing form gives the same Cartan type: it is an invariant positive form on the compact algebra, and on a simple factor its comparison with the negative intrinsic Killing form is scalar. Indeed the comparison is self-adjoint and commutes with every adjoint map; each eigenspace is an ideal, so simplicity forces a single eigenvalue. Orthogonal factors have no cross pairing because they commute and each equals its derived algebra. Thus the Cartan matrix read from (AH.18) is the intrinsic one. Finally AC.1 and AA.3 identify the compact form as \(\mathfrak{sp}(4)\). Its dimension is \(4+32=36\), as asserted. □

**Theorem AH.3 (exhaustion of the \(E_6\) diagram-reversal signs).** All signed representatives of AB.2 with permutation (AH.1) belong to exactly two compact-conjugacy classes. The all-positive choice has fixed algebra \(\mathfrak f_4^c\); every other choice has fixed algebra \(\mathfrak{sp}(4)\). Together with the two nonidentity toral classes of AG.1 and the identity, these are all compact involution classes of \(\mathfrak e_6^c\).

**Proof.** Only vertices 2 and 4 can carry signs in AB.2; paired vertices have coefficient \(+1\). Write a representative as \(d_b\Gamma\), with sign bits \((b_2,b_4)\in\{0,1\}^2\). If \(i\) is fixed by \(\pi\), then \(\Gamma\) fixes \(e_i-f_i\) and therefore commutes with the compact reflection
\[
n_i=\exp\bigl(\tfrac{\pi}{2}\operatorname{ad}(e_i-f_i)\bigr)
\]
of AC.3. Hence
\[
n_i(d_b\Gamma)n_i^{-1}
=(n_id_bn_i^{-1})\Gamma.
\tag{AH.19}
\]
The toral factor changes by the proved row-sign formula (AC.12).

At black vertex 2, only its neighbour 4 changes sign. Thus \((1,0)\) and \((1,1)\) are conjugate. At black vertex 4, its neighbours 2, 3 and 5 change sign. This toggles \(b_2\), while leaving \(b_4=1\) and temporarily putting equal negative signs at the paired vertices 3 and 5.

Those paired signs can be removed by compact conjugation, not by discarding coefficients. Let \(q\) be the toral compact automorphism with scalar \(q_5=-1\) and all other simple-generator scalars \(q_j=1\); AC.3 supplies it. If an automorphism sends \(e_j\) to \(c_j e_{\pi(j)}\), conjugation by \(q\) changes the coefficient to
\[
c_j\,\frac{q_{\pi(j)}}{q_j}.
\tag{AH.20}
\]
Both negative coefficients at 3 and 5 become positive; coefficients at fixed vertices and at 1, 6 remain unchanged. The inverse coefficients on \(f_j\) change consistently because \(q\) is an automorphism. This gives a compact conjugacy between \((0,1)\) and \((1,1)\). The three nonzero bit pairs are therefore one class. It contains \((1,0)\), whose fixed algebra is \(\mathfrak{sp}(4)\) by AH.2.

The zero pair is \(\Gamma\), with fixed dimension 52 by AH.1, whereas the nonzero class has fixed dimension 36. They are not conjugate. The toral nonidentity classes have fixed dimensions 46 and 38 by AF.2; the identity has fixed dimension 78. These five numbers are distinct, so no two of the five displayed classes merge under any compact automorphism.

Finally the \(E_6\) diagram has a unique valence-three vertex and legs of lengths \(1,2,2\). An automorphism fixes the node and the length-one leg, and either fixes or exchanges the equal length-two legs. These are exactly the identity and (AH.1). AB.2 says that every compact involution has a representative for one of these permutations. AF.3 exhausts the identity case and the preceding four-sign calculation exhausts (AH.1), proving the full assertion. □

## AI. Algebraic exhaustion and the Hermitian sublist

**Theorem AI.1 (the full algebraic list).** The twelve exceptional nonidentity compact-involution classes consist of the ten rows of AG.1 and the following two rows:

| Family | \(\mathfrak u\) | \(\mathfrak k\) | Real dual | \(\dim\mathfrak p\) |
| --- | --- | --- | --- | ---: |
| EI | \(\mathfrak e_6^c\) | \(\mathfrak{sp}(4)\) | \(\mathfrak e_{6(6)}\) | 42 |
| EIV | \(\mathfrak e_6^c\) | \(\mathfrak f_4^c\) | \(\mathfrak e_{6(-26)}\) | 26 |

Every effective irreducible compact semisimple symmetric Lie-algebra pair occurs in one of these lists:

1. A classical pair in AE.2, with its stated flat and product exceptions removed.
2. One of the twelve exceptional pairs in AG.1 and the table above.
3. A group-type pair \((\mathfrak v\oplus\mathfrak v,\operatorname{diag}\mathfrak v)\), where \(\mathfrak v\) is compact simple of any of the classical or exceptional types.

The converse holds for all entries in these ranges. Their noncompact duals are obtained by replacing the odd space \(\mathfrak p\) by \(i\mathfrak p\); in the group-type case the dual is \((\mathfrak v_{\mathbb C},\mathfrak v)\), with the first algebra regarded as real.

The nonflat irreducible Hermitian algebraic pairs are precisely AIII, CI, DIII, BDI with one parameter two and the other different from two, EIII and EVII, together with their real duals. AI at rank parameter two repeats the rank-one entry, as explained in AE.2. The lists give exhaustion; they may contain further identifications at small ranks.

**Proof.** AH.1–AH.3 give the two additional \(E_6\) classes and distinguish them from every toral class. Their fixed dimensions are 36 and 52; subtracting from 78 gives odd dimensions 42 and 26. Formula (AG.1) gives the Killing signature differences
\[
42-36=6,\qquad26-52=-26,
\tag{AI.1}
\]
which are exactly the real-dual subscripts in the table. The pairs are effective and irreducible by AE.1, and AA.5 constructs and distinguishes the corresponding real forms. AG.1 already exhausts \(G_2,F_4,E_7,E_8\), while AH.3 exhausts \(E_6\). Thus these are all twelve exceptional nonidentity involution classes, rather than a collection of examples.

For general exhaustion, AE.1 reduces any effective irreducible compact semisimple pair either to a nonidentity involution of a compact simple algebra or to a two-factor exchange. The same theorem proves that the complexification of a compact simple algebra is complex simple. U.1 exhausts its complex types, and AA.1–AA.3 give existence and uniqueness of their compact forms. For the classical types, AD.1–AD.3 give every involution and AE.2 gives the pairs, dimensions and boundary cases. For the exceptional types, AG.1 and AH.3 give every involution. A two-factor exchange is precisely the group-type pair in item 3, whose effectiveness, irreducibility and displayed real dual were all proved in AE.1. This establishes both exhaustion and the converse, including all simple types.

D.3 proves that duality preserves effectiveness and the odd isotropy representation and exchanges the two Killing-sign types. Thus it preserves irreducibility and gives the stated noncompact pairs with their invariant metrics. Conversely, for an effective irreducible noncompact semisimple orthogonal pair, its dual has compact type by that theorem, so applying the compact exhaustion to the dual includes the original pair after dualizing again. No classification of matrix groups or global quotient topology is used here.

For the Hermitian sublist, AE.2 already gives exactly the classical cases and excludes the flat BDI \((1,1)\) and the reducible BDI \((2,2)\) boundaries. AG.1 gives EIII and EVII among its ten exceptional rows. The two additional rows have compact simple fixed algebras \(\mathfrak{sp}(4)\) and \(\mathfrak f_4^c\); AC.1 and AE.1 give zero centres. E.2 therefore excludes both from the Hermitian list. Group-type isotropy is the centre-free simple algebra \(\mathfrak v\), so E.2 excludes it too. That theorem also proves that every listed Hermitian pair has exactly two infinitesimally invariant complex structures, opposite in sign, compatible with every positive scale of its invariant metric; D.3 transfers them to the real duals. For any global presentation with disconnected isotropy, descent still requires the full stabilizer condition A.3. This completes the algebraic assertions without presuming the global homogeneous-space identifications. □

## AJ. From the algebraic list to complete geometry

For a symmetric pair, the **metric data** mean a positive inner product \(b\) on its odd space, invariant under the even algebra. An isomorphism of metric pairs is a Lie-algebra isomorphism which commutes with the involutions and is an isometry on the odd spaces. This retains the scale of each factor.

**Theorem AJ.1 (the simply connected realization and its uniqueness).** Let \((\mathfrak g,\theta)\) be an effective semisimple orthogonal symmetric pair, with even and odd spaces \(\mathfrak h,\mathfrak m\), and metric data \(b\). There is a complete simply connected Riemannian symmetric space \(X(\mathfrak g,\theta,b)\) realizing these data. It has the presentation
\[
X(\mathfrak g,\theta,b)=\widetilde G/H,
\qquad H=(\widetilde G^\sigma)^0,
\tag{AJ.1}
\]
where \(\widetilde G\) is a simply connected group with algebra \(\mathfrak g\), and \(\sigma\) integrates \(\theta\). The subgroup \(H\) is closed and connected.

An isomorphism of metric pairs gives a unique pointed isometry of their realizations with its prescribed odd differential. Conversely, every pointed isometry gives such a metric-pair isomorphism. For any connected Riemannian symmetric presentation of these same data, allowing disconnected isotropy, its Riemannian universal cover is this realization. In particular the simply connected model is independent of the chosen group presentation.

**Proof.** C.4 constructs the connected group \(G_0=\operatorname{Aut}(\mathfrak g)^0\) with Lie algebra identified faithfully with \(\mathfrak g\). F.1 supplies its connected simply connected covering group \(\widetilde G\), with the same algebra. [Symmetric spaces C.1](symmetric-spaces.md#lemma-c-1) integrates \(\theta\) to an automorphism \(\sigma\) of \(\widetilde G\); uniqueness makes \(\sigma^2=1\).

The fixed set \(\widetilde G^\sigma\) is a subgroup and is closed: it is the equalizer of the continuous maps \(\sigma\) and the identity into a Hausdorff space. Invariant connections A.1 therefore gives it an embedded Lie-group structure. Its Lie algebra is precisely \(\mathfrak h\). One inclusion follows by differentiating \(\sigma(g)=g\); the other follows from
\[
\sigma(\exp tA)=\exp(t\theta A)=\exp(tA)
\qquad(A\in\mathfrak h).
\]
The identity component \(H\) is closed as well. Indeed the closure of a connected set is connected, so the closure of a component is contained in that same maximal connected component. It is connected by definition and has the same Lie algebra.

Every \(\operatorname{ad}A|_{\mathfrak m}\), \(A\in\mathfrak h\), is skew for \(b\). Differentiation of \(b(e^{t\operatorname{ad}A}X,e^{t\operatorname{ad}A}Y)\) shows that it is constant in \(t\). Exponentials generate connected \(H\), by [Flat connections A.1](flat-connections-and-infinitesimal-holonomy.md#lemma-a-1). Thus \(b\) is invariant under its full action. [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) constructs the smooth Hausdorff second-countable quotient in (AJ.1), and [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) gives its complete canonical Levi-Civita connection, invariant metric and global point symmetries. Its curvature at the identity coset is
\[
R_b(X,Y)Z=-[[X,Y],Z].
\tag{AJ.2}
\]
I.1 proves that the quotient is simply connected. This proves existence using an actual group and closed subgroup.

Suppose \(\varphi\) is an isomorphism of two metric pairs. Its odd restriction \(F\) preserves (AJ.2), as follows by applying \(\varphi\) to the three brackets, and preserves the tangent metric. Both constructed spaces have zero torsion, parallel curvature and complete connections. [Killing fields C.1](holonomy-killing-fields-and-analytic-extension.md#theorem-c-1) therefore extends \(F\) to a unique global affine isomorphism taking the specified base point to the specified base point. It is an isometry: the difference between the pulled-back target metric and the source metric is parallel by [Linear connections C.1](linear-and-affine-connections.md#theorem-c-1), vanishes at the base point, and hence vanishes along every path in the connected source. Uniqueness also follows from [Affine transformations A.1](affine-transformations-and-the-isometry-group.md#lemma-a-1).

For the converse, first note that
\[
\mathfrak h=[\mathfrak m,\mathfrak m].
\tag{AJ.3}
\]
Indeed C.2 splits an effective orthogonal pair into its Euclidean and semisimple symmetric ideals, and proves this equality on every semisimple summand. Here there is no nonzero Euclidean ideal: its translation space would be an abelian ideal of a semisimple algebra; if that space were zero, effectiveness would make its purely even part zero as well.

By C.1 the even representation is faithful. Equations (AJ.2)–(AJ.3) consequently identify the pair with its curvature algebra:
\[
\begin{aligned}
\mathfrak h&\longrightarrow
 \operatorname{span}\{R_b(X,Y)\}\subset\operatorname{End}(\mathfrak m),
 & A&\longmapsto\operatorname{ad}A|_{\mathfrak m},\\
[X,Y]&\longmapsto-R_b(X,Y).
\end{aligned}
\tag{AJ.4}
\]
The first map is onto because of (AJ.3), and is injective by faithfulness. Its compatibility with every bracket is the adjoint representation identity; the odd-odd bracket has the sign explicitly shown in (AJ.4). This is also the curvature-algebra construction proved in [Symmetric spaces C.2](symmetric-spaces.md#proposition-c-2).

The differential \(F\) of a pointed isometry preserves curvature, since it preserves the Levi-Civita connection by [Riemannian connections A.3](riemannian-connections-and-convex-neighbourhoods.md#theorem-a-3). On the curvature algebra it induces \(A\mapsto FAF^{-1}\), and on the odd space it induces \(F\). These maps preserve even commutators and the even action. They preserve the odd bracket because \(FR_b(X,Y)F^{-1}=R_{b'}(FX,FY)\). Through (AJ.4) they give the required metric-pair isomorphism. Its even part is forced by its odd part, so no extra choice is involved.

Finally take any connected Riemannian symmetric presentation of the same data. Its metric is complete by [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2). [Flat connections D.2](flat-connections-and-infinitesimal-holonomy.md#theorem-d-2) constructs its smooth universal cover; the lifted metric is complete by Hopf–Rinow E.4. Local isometries preserve the connection and curvature, so this cover has the same tangent metric, zero torsion and parallel curvature. [Killing fields C.1](holonomy-killing-fields-and-analytic-extension.md#theorem-c-1) and the parallel-metric argument above identify it isometrically with (AJ.1). This proves the covering and presentation assertions, including the possibility of disconnected isotropy downstairs. □

**Theorem AJ.2 (geometric exhaustion).** Every connected complete simply connected Riemannian manifold with parallel curvature is isometric to a product of a Euclidean space and finitely many simply connected irreducible spaces belonging to the classical, exceptional and group-type pairs of AI.1 or their noncompact duals. On each nonflat factor the metric is a positive multiple of the indicated Killing metric. Conversely every such finite product is a complete simply connected Riemannian symmetric space. The Euclidean dimension and nonflat metric factors are unique up to permutation, with repeated entries in the algebraic lists identified when their metric pairs are isomorphic.

Each compact-type factor is compact. Each noncompact-type factor has a global exponential diffeomorphism from its tangent space. Thus the algebraic tables exhaust the simply connected geometry, even before particular matrix-quotient names or small-rank repetitions are resolved.

**Proof.** Fix \(o\in M\), write \(V=T_oM\), and form the curvature algebra of [Symmetric spaces C.2](symmetric-spaces.md#proposition-c-2):
\[
\mathfrak a=\mathfrak h_R\oplus V,\qquad
\mathfrak h_R=\operatorname{span}\{R_o(X,Y):X,Y\in V\}.
\tag{AJ.5}
\]
Its even part acts on \(V\) by its given endomorphisms, and \([X,Y]=-R_o(X,Y)\). That theorem proves all Jacobi identities and the involution, so (AJ.5) is a Lie algebra, not only a formal bracket prescription.

We verify the orthogonal hypothesis needed for C.2. Curvature operators are skew-adjoint for the tangent metric \(b=g_o\): the induced curvature acting on the parallel metric is zero, and the tensor rule of [Linear connections C.1](linear-and-affine-connections.md#theorem-c-1) expands this as
\[
0=-b(R(X,Y)U,W)-b(U,R(X,Y)W).
\]
On \(\mathfrak h_R\) set \(Q(A,B)=-\operatorname{tr}(AB)\). For a skew-adjoint \(A\),
\[
Q(A,A)=\operatorname{tr}(A^*A),
\]
which is positive unless \(A=0\); the sum of squared matrix entries proves this directly. Cyclicity of trace gives
\[
Q([D,A],B)+Q(A,[D,B])=0
\qquad(D,A,B\in\mathfrak h_R).
\tag{AJ.6}
\]
Indeed the sum is the negative trace of \(DAB-ABD\). Hence \(Q\oplus b\), with the two summands orthogonal, is a positive form on \(\mathfrak a\) for which every even adjoint action is skew. The even action is faithful, since \(\mathfrak h_R\) is a space of actual endomorphisms; C.1 makes the pair effective. Its odd space generates it, because odd brackets span \(\mathfrak h_R\).

C.2 now splits (AJ.5) into an abelian translation ideal and effective semisimple symmetric ideals of compact or noncompact type, each with irreducible nonzero odd representation. The splitting is orthogonal for the particular \(b\) just used in that theorem. Distinct ideals commute. The curvature formula therefore has no mixed terms and on each odd summand is precisely its own double-bracket curvature.

Construct the simply connected space for each semisimple ideal by AJ.1, using the restricted metric, and use Euclidean space with the restricted constant metric for the translation ideal. Let \(N\) be their finite metric product. It is connected and simply connected: paths and contractions can be taken in each coordinate. It is complete because its geodesic equation is the separate equation on each factor. This product-connection formula follows from the Koszul formula: lifted vector fields from different factors commute, the metric pairs them to zero, and each factor's metric is constant along the other factors. The same computation shows that its curvature is the direct sum of the factor curvatures and that it is parallel.

At the product base point, the natural map \(T_oM\to T_{o'}N\) preserves the metric and all curvature components, by the preceding decomposition. [Killing fields C.1](holonomy-killing-fields-and-analytic-extension.md#theorem-c-1) gives a global affine isomorphism \(M\to N\), since both spaces are complete and simply connected. The parallel-metric argument in AJ.1 makes it an isometry. Thus no global product or existence assertion has been inferred merely from a table.

We also check irreducibility of the constructed nonflat factors as Riemannian spaces. If one split into a product of two positive-dimensional Riemannian manifolds, the product curvature formula would express its curvature algebra as a direct sum acting separately on the two tangent spaces. Each tangent space would then be invariant under the even algebra in (AJ.4), contradicting its irreducible odd representation. Conversely the preceding construction splits every pair having more than one nonzero symmetric odd summand. This identifies the algebraic irreducibility condition with the geometric one. The factors are nonflat because D.1 gives definite nonzero Ricci curvature.

D.3 turns each noncompact-type ideal into a compact-type ideal. AI.1 exhausts every compact irreducible pair and gives all its real duals. It therefore places every factor just constructed in the asserted list. D.1 supplies exactly the positive scalar multiples of the Killing metric on each factor; it also excludes cross terms between distinct nonflat summands. D.2 proves compactness for the compact-type factors and the global exponential conclusion for the noncompact-type factors.

For the converse, every listed algebraic pair is effective, irreducible and orthogonal by AI.1. AJ.1 realizes it, and the product of its point symmetries with Euclidean point reflections is a point symmetry on the product. The preceding product calculation proves completeness and simple connectivity. Finally [De Rham decomposition E.3](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-e-3) gives uniqueness of the Euclidean dimension and the nonflat metric factors. Within one such factor, AJ.1 says exactly when two algebraic entries give the same pointed metric geometry. Unpointed isometries give the same equivalence, since the spaces are homogeneous and an isometry can be followed by a target isometry carrying its image base point to the selected base point. This proves every exhaustion and uniqueness assertion. □

**Theorem AJ.3 (the complete simply connected Hermitian list).** A complete simply connected Hermitian symmetric space is a product of a flat complex Euclidean space and irreducible factors from AIII, CI, DIII, the nonflat irreducible BDI entries with one parameter two, EIII and EVII, or their noncompact duals. The low-rank repetition AI at parameter two is included by these entries. Each nonflat factor admits the two opposite invariant complex structures of E.2 and any positive metric scale. Each choice gives a global Hermitian symmetric structure on the simply connected product.

For a fixed Riemannian product decomposition, every compatible Hermitian symmetric complex structure preserves each nonflat factor and the Euclidean factor. It is one of the two stated choices on each nonflat factor, and a constant orthogonal complex structure on the Euclidean factor. The latter forces the Euclidean dimension to be even. These statements concern simply connected spaces; descent to any further quotient requires preservation of the complex structure.

**Proof.** A.2 makes the complex structure \(J\) of a Hermitian symmetric space parallel. Apply AJ.2 to its underlying Riemannian metric. The curvature action on a parallel endomorphism vanishes, by [Linear connections C.1](linear-and-affine-connections.md#theorem-c-1), so
\[
[R(X,Y),J]=0.
\tag{AJ.7}
\]
At the product base point, write the tangent decomposition as
\[
V=V_0\perp V_1\perp\cdots\perp V_s,
\]
where \(V_0\) is the Euclidean part. Its curvature algebra is
\(\mathfrak h_1\oplus\cdots\oplus\mathfrak h_s\), with \(\mathfrak h_a\) acting only on \(V_a\). On a nonflat irreducible factor,
\[
[\mathfrak h_a,V_a]=V_a.
\tag{AJ.8}
\]
The left side is invariant; irreducibility makes it either zero or all of \(V_a\). If it were zero, (AJ.2) would give zero curvature, contradicting the definite Ricci tensor of D.1.

For \(A\in\mathfrak h_a\) and \(v\in V_a\), equation (AJ.7) says \(J(Av)=A(Jv)\), whose right side lies in \(V_a\). Equation (AJ.8) therefore gives \(J(V_a)\subseteq V_a\). Also \(V_0\) is exactly the common fixed space of all the \(\mathfrak h_a\): an irreducible nontrivial \(V_a\) has no fixed vectors. Commutation then gives \(J(V_0)\subseteq V_0\). Parallelism extends these statements from the base point to the product distributions.

On each nonflat factor, \(J|_{V_a}\) is an invariant orthogonal complex structure. E.2 and AI.1 give exactly the displayed family list and exactly the two opposite choices. In particular, both exceptional Hermitian factors occur as actual complete simply connected spaces by AJ.1, with complex dimensions 16 and 27 from AG.1. The other exceptional and the group-type factors are excluded by their centre-free even algebras.

On the Euclidean factor parallelism means that the matrix of \(J\) in constant coordinates is constant. Choose a unit vector \(v\). Orthogonality and \(J^2=-1\) give \(\langle v,Jv\rangle=0\) and \(|Jv|=1\). The plane spanned by \(v,Jv\) is \(J\)-stable, and its orthogonal complement is also stable: pairing \(Jw\) with a vector in the plane is the negative pairing of \(w\) with its \(J\)-image. Induction gives an orthonormal basis of such pairs. Hence its dimension is \(2d\), and these coordinates identify it isometrically and complex linearly with \(\mathbb C^d\).

Conversely choose the indicated algebraic complex structure on each listed factor. The isotropy \(H\) in (AJ.1) is connected, so infinitesimal invariance gives full \(H\)-invariance by exponentiation. A.3 constructs its global integrable complex structure and proves that the metric is Kähler and Hermitian symmetric. The Euclidean factor has its constant complex structure, and product complex charts give the product structure. It is parallel, orthogonal and preserved by the product point symmetries. All positive scales are allowed by D.1 and E.2.

Finally these assertions are compatible with the isometries used in AJ.2. Two parallel endomorphisms agreeing at the base point agree along every path; an affine isometry whose base differential intertwines the chosen complex structures therefore intertwines them everywhere. Such a smooth map is holomorphic by the complex-coordinate criterion used in A.2. For a further quotient, the structure descends precisely when every identifying transformation preserves it; in a symmetric presentation this is the full stabilizer condition A.3. Thus no disconnected-isotropy descent is implicit in the simply connected list. □

## AK. Special-unitary and oriented real models

The graph convention remains \(E_Z=\{(v,Zv)\}\), with \(Z\) of size \(q\times p\) and \(p,q\geq1\). Write
\[
\begin{aligned}
H_{\mathbb C}&=S(U(p)\times U(q))\\
&=\{\operatorname{diag}(a,d):a\in U(p),\ d\in U(q),\
                         \det a\,\det d=1\}.
\end{aligned}
\tag{AK.1}
\]

**Theorem AK.1 (The special-unitary quotient models).** The two homogeneous spaces
\[
SU(p+q)/H_{\mathbb C},\qquad SU(p,q)/H_{\mathbb C},
\quad SU(p,q)=U(p,q)\cap SL(p+q,\mathbb C),
\tag{AK.2}
\]
are respectively the complex Grassmannian of L.1 and the matrix ball of L.2, with precisely their complex structures and metrics. Both acting groups and the full stabilizer \(H_{\mathbb C}\) are connected. The graph representatives \(k_Z\) and \(g_Z\) in (L.4) and (L.14) have determinant one. The compact model is simply connected, as is the noncompact model. Their effective symmetric algebras are the compact and noncompact AIII pair, with the opposite brackets and metric normalization of L.3.

**Proof.** Here and below a block diagonal matrix is identified with its two diagonal blocks. First we verify the determinant assertion without taking a branch of a complex root. For either sign \(\epsilon=1,-1\), put
\[
P_\epsilon=I+\epsilon Z^*Z,\qquad Q_\epsilon=I+\epsilon ZZ^*,
\qquad
B_\epsilon=
\begin{pmatrix}I&-\epsilon Z^*\\ Z&I\end{pmatrix}.
\tag{AK.3}
\]
For the minus sign \(Z\) is in the ball. Multiplication on the left by
\(\begin{pmatrix}I&0\\-Z&I\end{pmatrix}\) gives the upper triangular block matrix with diagonal blocks \(I,Q_\epsilon\). The multiplying matrix has determinant one, so \(\det B_\epsilon=\det Q_\epsilon\). These determinant rules follow directly from the alternating multilinear determinant: a lower block row addition preserves it, and an upper triangular block matrix has the product of the diagonal determinants. By K.1, the positive spectra of \(Z^*Z\) and \(ZZ^*\), with multiplicities, agree; their remaining eigenvalues are zero. Consequently
\(\det P_\epsilon=\det Q_\epsilon>0\). Multiplying \(B_\epsilon\) on the right by
\(\operatorname{diag}(P_\epsilon^{-1/2},Q_\epsilon^{-1/2})\) gives exactly \(k_Z\) or \(g_Z\). Thus
\[
\det k_Z=\det g_Z
=\frac{\det Q_\epsilon}
       {\sqrt{\det P_\epsilon}\sqrt{\det Q_\epsilon}}=1,
\tag{AK.4}
\]
where each square root is positive, by the positive eigenvalues in K.1.

For the compact action, L.1 supplies a unitary matrix \(u\) sending the reference plane \(E\) to any chosen plane. Choose a block diagonal unitary matrix \(h\) with \(\det h=(\det u)^{-1}\): one entry in its first block can have this value and all other entries can be one. Then \(uh\in SU(p+q)\) has the same image plane. This proves transitivity. The full stabilizer is the intersection of the full stabilizer in L.1 with \(SU(p+q)\), namely (AK.1). For the noncompact action the determinant-one section \(g_Z\) already proves transitivity, and the stabilizer calculation in L.2 gives exactly the same intersection. These groups are closed matrix Lie subgroups by Invariant connections A.1, and [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) gives their smooth quotients. The graph sections just proved to lie in the smaller groups show that the quotient identifications and their inverses are smooth. In the compact case translate these graph sections by fixed elements of \(SU(p+q)\) to cover the space. They give the same holomorphic graph transitions as L.1. In the noncompact case the section is global. Thus the complex atlases and the metrics are exactly those already constructed in L.1 and L.2.

The ball can also be described intrinsically as the set of positive \(p\)-planes for the form \(\operatorname{diag}(I_p,-I_q)\). The first projection is injective on any such plane: a nonzero vector in its kernel would have negative norm. Equal dimensions make the projection an isomorphism, so the plane is a unique graph \(E_Z\); its form is \(v^*(I-Z^*Z)v\), which is positive exactly for \(Z\) in the ball. Changing the sign of the preserved form makes the same planes negative and leaves the preserving group unchanged.

To check connectedness of the full stabilizer, diagonalize \(a\) and \(d\) separately by the commuting-Hermitian argument in K.1. List the eigenangles of both blocks as \(t_1,\ldots,t_{p+q}\). Their sum is \(2\pi k\) for an integer \(k\), by the determinant-one condition. Subtract \(2\pi k\) from one angle. This changes neither endpoint matrix and makes the total sum zero. In the two fixed diagonalizing bases, replace every angle by \(st_j\), \(0\leq s\leq1\). The two block paths have product determinant
\(\exp(is\sum_jt_j)=1\), and give a path in \(H_{\mathbb C}\) from the identity to \((a,d)\). The compact acting group is connected by K.1. For any \(g\in SU(p,q)\), let \(Z=g\cdot0\). Then \(g=g_Zh\) with \(h\in H_{\mathbb C}\), and \(s\mapsto g_{sZ}\), followed by a translated path in \(H_{\mathbb C}\), joins the identity to \(g\).

The involution induced by \(\operatorname{diag}(I_p,-I_q)\) preserves each special-unitary group and has full fixed subgroup (AK.1). Its odd matrices are exactly (L.18); the even algebra is the trace-zero block diagonal algebra. Thus L.3 proves the asserted brackets, curvature and common tangent metric. This even action is faithful: if \(dZ=Za\) for every rectangular \(Z\), the matrix units first imply that \(a,d\) are scalar and then that their scalars agree. Total trace zero makes that scalar zero. The odd space generates the whole special-unitary symmetric algebra: brackets of matrix units give the off-diagonal entries in each diagonal block, their diagonal differences and the remaining trace-zero scalar combination. Alternatively this is the effective irreducible AIII pair already proved in AD.1 and AE.2. AC.1 proves that the compact special-unitary algebra has negative-definite Killing form. Hence the compact Hermitian space of L.1 is of compact type, and I.2 proves its simple connectivity. For the noncompact space the convex contraction in L.2 already proves it. These conclusions concern the actual quotients (AK.2), with their full stabilizers. □

Let \(\mathrm{Gr}^{+}_p(\mathbb R^{p+q})\) denote the space of real \(p\)-planes together with an orientation of the plane. An orientation of a real vector space means a choice of the ordered bases whose changes of basis have positive determinant.

**Theorem AK.2 (Oriented Grassmannians and real matrix balls).** There are smooth homogeneous identifications
\[
\begin{aligned}
\mathrm{Gr}^{+}_p(\mathbb R^{p+q})
 &=SO(p+q)/(SO(p)\times SO(q)),\\
\mathcal D^{\mathbb R}_{q,p}
 &=SO_0(p,q)/(SO(p)\times SO(q)).
\end{aligned}
\tag{AK.5}
\]
where
\[
\mathcal D^{\mathbb R}_{q,p}
=\{Z\in M_{q,p}(\mathbb R):I-Z^TZ>0\}
\tag{AK.6}
\]
and \(SO_0(p,q)\) is the identity component of the determinant-one group preserving \(\operatorname{diag}(I_p,-I_q)\). Both full stabilizers in (AK.5) are connected. In graph coordinates the metrics are
\[
b^\epsilon_Z(U,V)
=\operatorname{tr}\bigl((I+\epsilon Z^TZ)^{-1}
             U^T(I+\epsilon ZZ^T)^{-1}V\bigr),
\qquad \epsilon=1,-1.
\tag{AK.7}
\]
They make the two spaces complete Riemannian symmetric spaces. The first is compact and connected; the second is bounded, convex and simply connected. Their curvature operators in graph tangent coordinates are
\[
R^\epsilon(Z,W)V
=\epsilon\bigl(ZW^TV-WZ^TV+VW^TZ-VZ^TW\bigr).
\tag{AK.8}
\]
The map forgetting orientation on the first space is a two-sheeted local isometry onto the unoriented Grassmannian. No simple-connectivity assertion about the compact quotient is implicit here.

**Proof.** Choose an oriented orthonormal basis of a plane and then an orthonormal basis of its perpendicular space so that the combined basis has the standard orientation of \(\mathbb R^{p+q}\). Gram–Schmidt here is the finite projection and normalization induction in K.1, over \(\mathbb R\); the sign of a last complement vector can be changed because \(q\geq1\). The resulting matrix lies in \(SO(p+q)\). This proves transitivity on oriented planes. A special orthogonal transformation preserving the oriented reference plane is block diagonal with its first block in \(SO(p)\); determinant one forces its second block into \(SO(q)\). Conversely every such block diagonal transformation stabilizes it. [Affine transformations G.1](affine-transformations-and-the-isometry-group.md#theorem-g-1) proves connectedness of each special orthogonal group by successive rotations in coordinate planes, including \(SO(1)=\{1\}\). Thus both this group and its stabilizer are connected. Closedness, compactness and the smooth quotient follow from Invariant connections A.1, [Local tools 0.1](local-tools-for-bundles-and-transport.md#0-analytic-and-linear-foundations) and [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup).

A graph carries the orientation obtained by projecting to the oriented first summand. The real version of \(k_Z\) in (L.4) has determinant one by (AK.4) and its first block \(P_+^{-1/2}\) has positive determinant. It therefore gives an oriented graph section. Conversely, in such a chart the coordinate is \(ca^{-1}\) and the orientation condition is \(\det a>0\); both are smooth conditions. Translated sections cover all oriented planes and justify the first smooth identification in (AK.5). Forgetting orientation has exactly two preimages at each plane. On the neighborhood of graphs over that plane the two choices give two disjoint copies of the same real matrix chart, so it is a two-sheeted covering.

For the second model all the graph identities in L.2 hold over \(\mathbb R\), with \(*\) replaced by \(T\). In particular the real \(g_Z\) of (L.14) preserves the form of signature \((p,q)\), has determinant one by (AK.4), and \(s\mapsto g_{sZ}\) is a path to it from the identity. It lies in \(SO_0(p,q)\). The action
\[
g\cdot Z=(c+dZ)(a+bZ)^{-1}
\tag{AK.9}
\]
is therefore transitive. To verify both its domain and its full stabilizer within this component, a nonzero vector \((v,Zv)\) has positive norm for the preserved form. Its image cannot have first component zero. Hence \(a+bZ\) is invertible, and its graph Gram matrix is
\[
I-W^TW=(a+bZ)^{-T}(I-Z^TZ)(a+bZ)^{-1}>0.
\]
Associativity of the graph action proves the action law. Fixing zero implies \(c=0\), and the form equations then imply \(b=0\), \(a\in O(p)\), \(d\in O(q)\).

For any form-preserving block matrix, \(a^Ta-c^Tc=I\), so \(a\) is invertible. Its determinant sign is continuous and hence constant on a connected component. On the identity component it is positive. The full stabilizer there consequently has \(\det a=1\), and the total determinant gives \(\det d=1\). Conversely \(SO(p)\times SO(q)\) lies in that component by the rotation paths above. This proves the exact stabilizer, including its connectedness. The identity component is a Lie subgroup: an exponential neighborhood of a matrix Lie group is connected and contains the identity, so its component is open; its other components are translates and therefore the identity component is also closed. [Local tools 4.1](local-tools-for-bundles-and-transport.md#4-quotients-by-a-closed-lie-subgroup) supplies the quotient. The map \(Z\mapsto g_Z(SO(p)\times SO(q))\) is a smooth global inverse to the orbit map.

The domain (AK.6) is the real operator-norm unit ball, by the real spectral proof in K.1. The vector triangle inequality proves its convexity and the finite-dimensional norm bounds prove boundedness and openness, exactly as in L.2. The contraction \(Z\mapsto sZ\) proves simple connectivity.

The graph-transition derivative and the two Gram-matrix identities (L.6)–(L.8), or (L.6), (L.13), (L.15), are identities of real matrices when the action matrix and \(Z\) are real. Thus they apply in every oriented real chart. Their cancellations prove that (AK.7) agrees on overlaps and is invariant. Positivity follows from
\[
b^\epsilon_Z(U,U)
=\bigl\|(I+\epsilon ZZ^T)^{-1/2}
 U(I+\epsilon Z^TZ)^{-1/2}\bigr\|_{\mathrm{HS}}^2.
\tag{AK.10}
\]
The metric on an oriented graph is independent of which orientation it has; it is the graph metric of [Symmetric spaces E.2](symmetric-spaces.md#example-e-2) on the underlying plane. The covering just constructed is therefore a local isometry.

For completeness and symmetry on the oriented space, the orthogonal map \(J=\operatorname{diag}(I_p,-I_q)\) induces an involution on oriented planes by pushing forward their oriented bases. It fixes the oriented plane \(E\) and is \(Z\mapsto-Z\) in its chart. It is an isometry of (AK.7): its tangent map is \(-I\), and the two Gram matrices do not change. It is global, even when \(\det J=-1\). Equivalently on (AK.5) it is induced by the automorphism \(g\mapsto JgJ\); this preserves the acting group and its stabilizer. The identical automorphism preserves \(SO_0(p,q)\) and acts as \(Z\mapsto-Z\) on its ball. Conjugation by the transitive actions gives global point symmetries at every point, and [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves completeness.

Finally the odd matrices are the real versions of (L.18); differentiation of the form equations and exponentiation as in L.3 proves this Lie-algebra identification. Their brackets are exactly the real versions of (L.19). The even action is \(V\mapsto dV-Va\), the metric at zero is \(\operatorname{tr}(U^TV)\), and [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) gives \(R=-[[\,\cdot,\cdot\,],\cdot]\). This proves (AK.8) with both signs. It is the same symmetric-algebra duality, including metric scale, as D.3 wherever the effective semisimple hypothesis is applicable. The direct construction also includes the low-dimensional flat case \(p=q=1\): its compact space is the circle of oriented lines in \(\mathbb R^2\), identified with the unit circle by the unique positively oriented unit vector on each line. □

## AL. Determinant potentials and the disc

For a smooth real function \(\phi\) on a complex coordinate domain, write \(\partial_V\phi\) for its complex-linear directional derivative and \(\bar\partial_U\) for the conjugate-linear one. More explicitly, these are the combinations \((\partial_x-i\partial_y)/2\) and \((\partial_x+i\partial_y)/2\) in each coordinate, contracted with \(V\) and \(\bar U\). Thus \(\partial_V Z=V\), \(\partial_V Z^*=0\), \(\bar\partial_U Z=0\), and \(\bar\partial_U Z^*=U^*\). We call \(\phi\) a potential for the convention \(g=\operatorname{Re}h\) precisely when
\(h(U,V)=\bar\partial_U\partial_V\phi\). This fixes the normalization.

**Proposition AL.1 (The two Grassmannian potentials).** On a compact graph chart and on the noncompact matrix ball, respectively, the metrics of L.1 and L.2 have potentials
\[
\phi_\epsilon(Z)=\epsilon\log\det(I+\epsilon Z^*Z),
\qquad \epsilon=1,-1.
\tag{AL.1}
\]
For a graph transition \(W=(c+dZ)(a+bZ)^{-1}\) induced by the corresponding unitary group, let \(L=a+bZ\). On its domain,
\[
\phi_\epsilon(W)=\phi_\epsilon(Z)-\epsilon\log|\det L|^2.
\tag{AL.2}
\]
The final term has zero mixed complex Hessian. In particular (AL.1) is a global real potential on the matrix ball; in the compact case it supplies compatible local potentials. Their fundamental two-forms are \(\omega=(i/2)\partial\bar\partial\phi_\epsilon\).

**Proof.** We first prove the determinant differential used in the calculation. In the permutation expansion of \(\det(I+tC)\), the constant coefficient is one. A term linear in \(t\) must use one entry of \(tC\) and identity entries in all other positions, hence must use the identity permutation and a diagonal entry \(C_{jj}\). Thus
\(\det(I+tC)=1+t\operatorname{tr}C+O(t^2)\).
Multiplicativity of the determinant now gives, for an invertible matrix \(A\),
\[
d(\det)_A(H)=\det A\,\operatorname{tr}(A^{-1}H),
\qquad
d(\log\det)_A(H)=\operatorname{tr}(A^{-1}H)
\tag{AL.3}
\]
on the positive Hermitian cone. The real logarithm, its derivative \(d\log x/dx=1/x\) for \(x>0\), and its identity \(\log(xy)=\log x+\log y\) are proved in [Local tools 0.5](local-tools-for-bundles-and-transport.md#lemma-0-5). Apply that derivative to the positive determinant for the second equality. The identity \(AA^{-1}=I\), differentiated, also proves
\[
d(A^{-1})_A(H)=-A^{-1}HA^{-1}.
\tag{AL.4}
\]
These formulas hold after complex-linear extension of the real directional derivatives.

Set \(P=I+\epsilon Z^*Z\) and \(Q=I+\epsilon ZZ^*\); both are positive by K.1. Since \(\epsilon^2=1\), (AL.3) gives
\[
\partial_V\phi_\epsilon=\operatorname{tr}(P^{-1}Z^*V).
\]
Applying \(\bar\partial_U\) and (AL.4), with all matrix factors in their stated order, gives
\[
\begin{aligned}
\bar\partial_U\partial_V\phi_\epsilon
&=\operatorname{tr}(P^{-1}U^*V)
-\epsilon\operatorname{tr}(P^{-1}U^*ZP^{-1}Z^*V)\\
&=\operatorname{tr}\bigl(P^{-1}U^*
                      (I-\epsilon ZP^{-1}Z^*)V\bigr)\\
&=\operatorname{tr}(P^{-1}U^*Q^{-1}V).
\end{aligned}
\tag{AL.5}
\]
For the last equality, \(QZ=ZP\); multiplying \(I-\epsilon ZP^{-1}Z^*\) on the left by \(Q\) gives
\(Q-\epsilon ZPP^{-1}Z^*=I\). This proves the inverse identity and then exactly the metrics (L.3) and (L.12), including their scale. Their positivity and Kähler property were proved in L.1 and L.2; nothing about a potential is being used as a substitute for those proofs.

For clarity this last two-form assertion is also a coordinate identity. In the scalar entries \(z_j\) of \(Z\), put \(c_{j\bar k}=\partial_{z_j}\partial_{\bar z_k}\phi_\epsilon\). Then \(h(U,V)=\sum c_{j\bar k}V_j\bar U_k\). The real two-form
\[
\omega=\frac{i}{2}\sum_{j,k}c_{j\bar k}\,dz_j\wedge d\bar z_k
\]
evaluated on \(U,V\) equals \((i/2)(h(V,U)-h(U,V))=\operatorname{Im}h(U,V)=g(iU,V)\). It is closed: in its exterior derivative the coefficients of two holomorphic coordinate differentials are symmetric in their holomorphic indices, by commutation of third derivatives of the smooth potential, while their wedge product is antisymmetric. The same argument cancels the terms with two antiholomorphic differentials. This proves the formula and closedness directly in the stated convention.

Equations (L.7) and (L.13) both state \(P(W)=L^{-*}P(Z)L^{-1}\). Taking positive determinants gives
\(\det P(W)=\det P(Z)/|\det L|^2\), and real logarithms give (AL.2). To verify the zero-Hessian assertion without assuming a holomorphic logarithm, let \(f=\det L\), a nonvanishing holomorphic function on this coordinate domain. Differentiating the real function \(\log(f\bar f)\) gives
\(\partial_V\log|f|^2=(\partial_Vf)/f\).
For constant coordinate \(V\) this is holomorphic, since \(f\ne0\); its conjugate-complex derivative is therefore zero. This proves the mixed-Hessian claim. The complex chain rule introduces no extra mixed term under a holomorphic coordinate change, because \(\bar\partial_U(dW(V))=0\). Thus the local Hessians transform as the already identified Hermitian metric. □

**Proposition AL.2 (The Siegel potential with its exact scale).** For the metric (M.10) on \(\mathcal H_r\), the global potential is
\[
\psi(W)=-4\log\det Y,\qquad Y=\frac{W-\bar W}{2i}.
\tag{AL.6}
\]
For \(W'=(AW+B)(CW+D)^{-1}\) and \(N=CW+D\),
\[
\psi(W')=\psi(W)+4\log|\det N|^2.
\tag{AL.7}
\]
Under the Cayley map \(W=i(I+Z)(I-Z)^{-1}\) on the symmetric matrix ball,
\[
\psi(W(Z))
=-4\log\det(I-Z^*Z)+4\log|\det(I-Z)|^2.
\tag{AL.8}
\]
Its mixed Hessian is therefore exactly the metric (M.18).

**Proof.** Work in the complex vector space of symmetric matrices, with constant symmetric tangent vectors \(U,V\). Positivity of \(Y\) and the positive real logarithm are justified by (M.1). The coordinate derivatives are
\[
\partial_VY=\frac{V}{2i},\qquad
\bar\partial_UY=-\frac{\bar U}{2i}.
\]
Using (AL.3)–(AL.4) yields
\[
\begin{aligned}
\partial_V\psi&=2i\,\operatorname{tr}(Y^{-1}V),\\
\bar\partial_U\partial_V\psi
&=2i\,\operatorname{tr}
\left(-Y^{-1}\left(-\frac{\bar U}{2i}\right)Y^{-1}V\right)
=\operatorname{tr}(Y^{-1}\bar UY^{-1}V).
\end{aligned}
\tag{AL.9}
\]
This is precisely (M.10). Treating the tangent matrices as symmetric throughout automatically counts each off-diagonal coordinate twice, as in (N.1); no independent-coordinate factor is missing.

The domain and invertibility of \(N\) were proved in M.1. Equation (M.7) says \(Y'=N^{-*}YN^{-1}\), so taking determinants gives (AL.7). Its extra term has zero mixed Hessian by the explicit holomorphic-function calculation in AL.1. The Cayley map is a biholomorphism by M.3, and (M.20), with \(A=I-Z\) and \(P=I-Z^*Z\), says \(Y=A^{-*}PA^{-1}\). Taking determinants and logarithms gives (AL.8). Finally restricting the minus-sign Hessian (AL.5) to the complex linear subspace of symmetric matrices introduces no derivative correction: a constant symmetric tangent is also a constant tangent in the ambient matrix space. The last term of (AL.8) again has zero mixed Hessian. The result is four times that restriction, exactly (M.18). This proves both transformation laws and the normalization. □

**Proposition AL.3 (Disc translations and reflections).** For \(a\) in the unit disc \(\mathbb D\), the map
\[
f_a(z)=\frac{z-a}{1-\bar a z}
\tag{AL.10}
\]
is a biholomorphic isometry for \(g_-=|dz|^2/(1-|z|^2)^2\), with inverse
\[
f_a^{-1}(w)=\frac{w+a}{1+\bar a w}.
\tag{AL.11}
\]
The point symmetry at \(a\) is
\[
s_a(z)=f_a^{-1}(-f_a(z))
=\frac{2a-(1+|a|^2)z}{1+|a|^2-2\bar a z}.
\tag{AL.12}
\]
These statements also hold for the metric \(4g_-\). Their Gaussian curvatures are respectively \(-4\) and \(-1\).

**Proof.** The denominators of (AL.10) and (AL.11) are nonzero on the disc because \(|\bar az|<1\) and \(|\bar aw|<1\). Expanding two squared absolute values gives
\[
|1-\bar az|^2-|z-a|^2=(1-|a|^2)(1-|z|^2),
\]
and hence
\[
1-|f_a(z)|^2
=\frac{(1-|a|^2)(1-|z|^2)}{|1-\bar az|^2},
\qquad
f_a'(z)=\frac{1-|a|^2}{(1-\bar az)^2}.
\tag{AL.13}
\]
The first identity proves that \(f_a\) maps the disc into itself. Replacing \(a\) by \(-a\) proves the same for (AL.11); direct substitution shows that they are inverses. Their nonzero denominators make both maps holomorphic. Substitution of (AL.13) into the metric gives
\[
\frac{|f_a'(z)|^2}{(1-|f_a(z)|^2)^2}
=\frac1{(1-|z|^2)^2},
\]
which proves the isometry assertion. The map \(z\mapsto-z\) is also an isometry, is an involution and has derivative \(-1\) at zero. Since \(f_a(a)=0\), conjugating this map proves that \(s_a\) is an involutive isometry fixing \(a\) with derivative \(-1\). Combining the two fractions gives exactly (AL.12). Its displayed denominator is nonzero as well: \(1+|a|^2-2|a||z|>(1-|a|)^2>0\). Multiplying a metric by four preserves its isometries. The curvature values, including their normalization, were proved in L.4. □

## AM. Complex quadrics with the induced metric

We use \(g_{\mathrm{FS}}\) for the projective metric obtained from L.1 with \(p=1\). Its holomorphic sectional curvature is \(4\), by L.3. A positive constant \(\alpha\) specifies the ambient metric \(\alpha g_{\mathrm{FS}}\). The bilinear expression \(z^Tz=\sum z_j^2\) below has no complex conjugation.

**Theorem AM.1 (The quadric and the oriented two-plane model).** For \(q\geq1\), the set
\[
Q^q=\{[z]\in\mathbb {CP}^{q+1}:z^Tz=0\}
\tag{AM.1}
\]
is a smooth complex hypersurface. The map
\[
\begin{aligned}
\Phi:\mathrm{Gr}^{+}_2(\mathbb R^{q+2})&\longrightarrow Q^q,\\
\operatorname{span}^{+}(u,v)&\longmapsto[u+iv],
\end{aligned}
\tag{AM.2}
\]
where \(u,v\) is any positively oriented orthonormal basis, is a diffeomorphism. It identifies \(Q^q\) with
\[
SO(q+2)/(SO(2)\times SO(q)).
\tag{AM.3}
\]
At the reference plane, write a real graph tangent as \(X=(x,y)\in M_{q,2}(\mathbb R)\). The induced metric and complex structure are exactly
\[
b_\alpha(X,Y)=\frac{\alpha}{2}\operatorname{tr}(X^TY),
\qquad J(x,y)=(-y,x).
\tag{AM.4}
\]
Globally the metric is \(\alpha/2\) times the compact real graph metric (AK.7). The quadric is compact, simply connected and complete Hermitian symmetric. At an oriented plane \(E\), its point symmetry is induced on projective space by the real orthogonal transformation that is \(+I\) on \(E\) and \(-I\) on \(E^\perp\).

**Proof.** Let \(z=a+ib\ne0\) with \(a,b\) real. The null equation is
\[
|a|^2=|b|^2,\qquad a\cdot b=0.
\tag{AM.5}
\]
Their common length is positive, so they determine an oriented two-plane. Multiplication of \(z\) by \(c+id\ne0\) changes its real and imaginary parts to
\((ca-db,da+cb)\); the corresponding change of ordered basis has determinant \(c^2+d^2>0\). Thus the oriented plane depends only on \([z]\). Conversely rotation of a positively oriented orthonormal basis changes \(u+iv\) by a unit complex scalar. This proves that (AM.2) is well defined and is a bijection, with the inverse just described.

Here are explicit holomorphic charts on the image. For a point \([z_0]\in Q^q\), choose a representative with \(z_0^*z_0=1\). Then
\(z_0^Tz_0=0\) and \(z_0^T\bar z_0=1\). Let \(W\) be the complexification of the real perpendicular space to the plane determined by \(z_0\). The vectors \(z_0,\bar z_0\) form a Hermitian orthonormal basis of the complexification of that plane; together with \(W\) they give a direct decomposition of \(\mathbb C^{q+2}\). On the projective chart where \(z^T\bar z_0\ne0\), normalize this coefficient to one and write
\[
z=z_0+w+t\bar z_0,\qquad w\in W,\quad t\in\mathbb C.
\]
The null equation is exactly \(2t+w^Tw=0\). Thus
\[
w\longmapsto
\left[z_0+w-\frac12(w^Tw)\bar z_0\right]
\tag{AM.6}
\]
is a holomorphic parametrization of the whole part of \(Q^q\) in that chart. Its inverse is the \(W\)-component after the indicated projective normalization, hence is holomorphic. In the ambient affine coordinates \((w,t)\), the image is the graph of the polynomial \(-w^Tw/2\). It is therefore an embedded complex hypersurface of dimension \(q\). These charts cover \(Q^q\), since each chosen \([z_0]\) lies in its own chart.

The smooth local orthonormal frames for oriented graphs constructed in AK.2 make \(\Phi\) smooth. In the opposite direction choose a smooth nonzero projective representative locally, split it into its real and imaginary parts and divide both by their common positive length from (AM.5). This is a smooth oriented orthonormal frame, hence gives a smooth map to the oriented Grassmannian. The two inverse maps are smooth. They intertwine the real \(SO(q+2)\) action and its complex-linear projective action. Transitivity and the full stabilizer in (AM.3) are consequently exactly those proved in AK.2; in particular the stabilizer is connected.

We spell out the projective metric normalization. For a nonzero representative \(z\) and tangent lifts \(v,w\in\mathbb C^{q+2}\), the projective Hermitian form is
\[
h^{\mathrm{FS}}_{[z]}([v],[w])
=\frac{(z^*z)(v^*w)-(v^*z)(z^*w)}{(z^*z)^2}.
\tag{AM.7}
\]
Here \([v]\) denotes the derivative of \(t\mapsto[z+tv]\), rather than a projective point. To verify the formula from L.1, make a unitary change of coordinates taking \(z/|z|\) to the first coordinate vector. At this point the affine graph tangent is the perpendicular component of \(v\) divided by \(|z|\); L.1 gives its ordinary Hermitian inner product. Removing the components along \(z\) gives exactly (AM.7). Unitary invariance in L.1 makes this calculation valid at every point, and adding multiples of \(z\) to tangent lifts has no effect on the expression.

At \(E=\operatorname{span}^{+}(e_1,e_2)\), the oriented graph section \(k_{tX}\) of AK.2 has first two columns with derivatives \(x,y\) in \(E^\perp\). Indeed its diagonal factor \((I+t^2X^TX)^{-1/2}\) has derivative zero at zero by K.1. The derivative of \(\Phi\) is therefore represented by \(x+iy\) at \(z=e_1+ie_2\). This lift is Hermitian orthogonal to \(z\), which has squared norm two. Taking the real part in \(\alpha\) times (AM.7) proves the metric in (AM.4). Multiplication of the lift by \(i\) sends \(x+iy\) to \(-y+ix\), proving its complex-structure formula. Both metrics are invariant under the transitive group, so equality at this one plane proves the asserted global factor \(\alpha/2\).

The indicated orthogonal reflection extends complex linearly to a unitary transformation; it preserves \(z^Tz\), hence \(Q^q\), and acts holomorphically and isometrically there. It fixes \([u+iv]\) and sends every tangent lift in \(E^\perp\otimes\mathbb C\) to its negative. Thus it is a global point symmetry. The induced metric is Hermitian by (AM.7), so A.2 proves it is Kähler, and [Symmetric spaces A.2](symmetric-spaces.md#theorem-a-2) proves completeness. Compactness and connectedness follow from AK.2.

For simple connectivity, the symmetric algebra is \(\mathfrak{so}(q+2)\) with even part \(\mathfrak{so}(2)\oplus\mathfrak{so}(q)\), as in AK.2. Its even action on \(X\) is \(dX-Xa\). If it vanishes on all matrix units, \(a,d\) must be the same real scalar matrices; skew symmetry then makes them zero. Thus the action is faithful, and C.1 makes the pair effective. Its Killing form is negative definite by AC.1 for every \(q+2\geq3\), including \(\mathfrak{so}(4)\). The space is therefore of compact type, and I.2 proves that this actual compact Hermitian symmetric quotient is simply connected.

We will also use a consequence of (AM.7) and AL.1. If \(F\) is a holomorphic nonzero vector-valued local lift of a map to projective space, its pulled-back Hermitian form has potential
\[
\log(F^*F).
\tag{AM.8}
\]
To prove this, choose an entry \(F_j\ne0\). The graph coordinates of the map are the other ratios \(F_k/F_j\), and
\[
\log(F^*F)=\log|F_j|^2+
\log\left(1+\sum_{k\ne j}|F_k/F_j|^2\right).
\]
The first term has zero mixed Hessian by AL.1. The second is precisely the projective potential of AL.1 pulled back by the holomorphic graph coordinates. The complex chain rule, with its vanishing mixed derivative of a holomorphic map, proves the assertion, including the normalization in (AM.7). □

**Theorem AM.2 (Ricci scale, irreducibility and the real dual).** With the metric induced by \(\alpha g_{\mathrm{FS}}\), the quadric has
\[
\operatorname{Ric}_{Q^q}=\frac{2q}{\alpha}\,g_{Q^q}.
\tag{AM.9}
\]
It is irreducible for \(q=1\) and for \(q\geq3\). Its noncompact dual is the real matrix ball
\[
\mathcal D^{\mathbb R}_{q,2}
=SO_0(2,q)/(SO(2)\times SO(q))
\tag{AM.10}
\]
with \(\alpha/2\) times the negative-sign metric (AK.7) and the invariant complex structure whose value at zero is \(J(x,y)=(-y,x)\). This is a complete simply connected Hermitian symmetric space, and
\[
\operatorname{Ric}_{\mathcal D^{\mathbb R}_{q,2}}
=-\frac{2q}{\alpha}\,g_{\mathcal D^{\mathbb R}_{q,2}}.
\tag{AM.11}
\]

**Proof.** We compute Ricci from the already proved curvature, rather than infer the constant from a family name. On \(M_{q,p}(\mathbb R)\), use the Frobenius inner product
\(\langle U,V\rangle_F=\operatorname{tr}(U^TV)\) and its matrix-unit orthonormal basis \(E_{aj}\). By (AK.8), the trace defining compact Ricci is
\[
\sum_{a=1}^q\sum_{j=1}^p
\left\langle
E_{aj}U^TV-UE_{aj}^TV+VU^TE_{aj}-VE_{aj}^TU,\
E_{aj}\right\rangle_F.
\tag{AM.12}
\]
The four sums can be evaluated entry by entry. The first summand has selected entry \((U^TV)_{jj}\), giving \(q\operatorname{tr}(U^TV)\) after summing. The second has selected entry \(-U_{aj}V_{aj}\), giving \(-\operatorname{tr}(U^TV)\). The third has selected entry \((VU^T)_{aa}\), giving \(p\operatorname{tr}(U^TV)\). The fourth has selected entry \(-V_{aj}U_{aj}\), again giving \(-\operatorname{tr}(U^TV)\). Thus the sum is
\[
(p+q-2)\operatorname{tr}(U^TV).
\tag{AM.13}
\]
Ricci is the trace of \(T\mapsto R(T,U)V\), as in [Sectional curvature A.1](sectional-curvature-and-space-forms.md#theorem-a-1), so the computation may use any basis. In particular constant rescaling of the metric does not change this covariant Ricci tensor. Setting \(p=2\) and comparing with (AM.4) gives (AM.9) at the reference point and hence everywhere by invariance. The dual curvature is its negative by AK.2; the same trace calculation gives (AM.11) for the same tangent metric.

For comparison with the Killing normalization, we also compute its coefficient directly. On \(\mathfrak{so}(N)\), \(N\geq2\), let \(F_{ij}=E_{ij}-E_{ji}\) for \(i<j\). The Killing form \(B(A,B)=\operatorname{tr}(\operatorname{ad}A\operatorname{ad}B)\) is invariant under conjugation by every orthogonal matrix, because conjugation intertwines the adjoint operators. For two distinct unordered index pairs, choose an index belonging to exactly one pair and change the sign of that coordinate. This conjugation negates exactly one of the corresponding \(F\)'s, so their Killing pairing is zero. Their ordinary trace pairing is also zero, by matrix multiplication.

For \(F_{12}\), the two-dimensional span of \(F_{1k},F_{2k}\), for each \(k>2\), is invariant under its adjoint action:
\[
[F_{12},F_{1k}]=-F_{2k},\qquad
[F_{12},F_{2k}]=F_{1k}.
\]
The square of that action is \(-I\). Its action on \(F_{12}\) and on all \(F_{k\ell}\) with \(k,\ell>2\) is zero. Hence \(B(F_{12},F_{12})=-2(N-2)\), whereas \(\operatorname{tr}(F_{12}^2)=-2\). Coordinate permutations give the same diagonal coefficient for every pair. Bilinearity and the vanishing cross pairings prove
\[
B(A,B)=(N-2)\operatorname{tr}(AB).
\tag{AM.15}
\]
In the compact odd block \(P_X=\begin{pmatrix}0&-X^T\\X&0\end{pmatrix}\), multiplication gives \(\operatorname{tr}(P_XP_Y)=-2\operatorname{tr}(X^TY)\). At \(N=q+2\), therefore, \(B(P_X,P_Y)=-2q\operatorname{tr}(X^TY)\), consistent with the Ricci calculation above.

It remains to justify the complex and global assertions for (AM.10). The full stabilizer \(SO(2)\times SO(q)\) acts by \(X\mapsto dXa^{-1}\). Put
\[
j=\begin{pmatrix}0&-1\\1&0\end{pmatrix};
\qquad JX=-Xj.
\tag{AM.14}
\]
Every element of \(SO(2)\) is \(\begin{pmatrix}c&-s\\s&c\end{pmatrix}\) with \(c^2+s^2=1\), as follows from its two oriented orthonormal columns. It commutes with \(j\). Thus (AM.14) commutes with the action of the entire stabilizer, squares to \(-I\), and preserves the positive tangent metric \(\alpha\langle\,\cdot,\cdot\,\rangle_F/2\). A.3 supplies actual complex coordinates, parallelism and the complete Hermitian symmetric structure; AK.2 supplies the simply connected convex global domain. This argument does not claim that the real entries of \(X\) themselves are holomorphic coordinates away from zero.

The effective symmetric pair in either case is BDI with parameters \(2,q\), and the exact opposite-bracket realization is AK.2. AE.2 proves its irreducibility when \(q\ne2\); geometric irreducibility follows by the curvature-algebra argument in AJ.2. More explicitly a nontrivial Riemannian product would split its tangent space into nonzero subspaces invariant under all curvature operators, hence under the effective even algebra generated by the odd brackets. That contradicts the irreducible isotropy representation proved in AE.2. □

## AN. The conic and the quadric surface

**Proposition AN.1 (The conic as a scaled projective line).** The map
\[
\begin{aligned}
\nu:\mathbb {CP}^1&\longrightarrow Q^1\subset\mathbb {CP}^2,\\
[s:t]&\longmapsto[s^2-t^2:i(s^2+t^2):2st]
\end{aligned}
\tag{AN.1}
\]
is a biholomorphism. If \(Q^1\) has the metric induced by \(\alpha g_{\mathrm{FS}}\), then
\[
\nu^*g_{Q^1}=2\alpha g_{\mathrm{FS}}.
\tag{AN.2}
\]
Consequently its Gaussian curvature is \(2/\alpha\).

**Proof.** The three homogeneous quadratic coordinates in (AN.1) do not vanish simultaneously: their first coordinate minus \(i\) times their second is \(2s^2\), and their first plus \(i\) times their second is \(-2t^2\). Homogeneity gives a well-defined holomorphic map on projective space. Direct expansion gives
\[
(s^2-t^2)^2+\bigl(i(s^2+t^2)\bigr)^2+(2st)^2=0,
\]
so it maps into \(Q^1\).

For the global inverse, put \(A=z_1-iz_2\), \(B=z_1+iz_2\), \(C=z_3\). The null equation is \(AB=-C^2\), and \(A,B\) cannot both be zero. On \(A\ne0\), set \([s:t]=[1:C/A]\). In these coordinates the image of (AN.1) has \((A,B,C)=(2,-2(t/s)^2,2(t/s))\); after multiplication by \(A/2\) this is the prescribed triple. On \(B\ne0\), set \([s:t]=[-C/B:1]\); the corresponding image triple is \((2(s/t)^2,-2,2(s/t))\), which becomes the prescribed one after multiplication by \(-B/2\). On the overlap the two ratios agree because \(AB=-C^2\). These are holomorphic inverse maps in projective charts. Thus \(\nu\) is a biholomorphism, including the points at either chart boundary.

The squared norm of the polynomial lift is
\[
|s^2-t^2|^2+|s^2+t^2|^2+4|st|^2
=2(|s|^2+|t|^2)^2.
\tag{AN.3}
\]
On the chart \(s\ne0\), take \(s=1\). The pullback potential (AM.8) for the ambient scale \(\alpha\) is
\(\alpha\log2+2\alpha\log(1+|t|^2)\).
Its mixed Hessian is \(2\alpha\) times the projective-line metric by AL.1. The same calculation on \(t=1\) covers the other chart, so it proves the global equality (AN.2). L.4 gives curvature \(4\) for \(g_{\mathrm{FS}}\) on \(\mathbb {CP}^1\) and proves that rescaling divides sectional curvature by the scale. Hence the curvature is \(4/(2\alpha)=2/\alpha\), agreeing with (AM.9) in real dimension two. □

**Proposition AN.2 (The quadric surface as an exact metric product).** The complex-linear map
\[
T:\mathbb C^4\longrightarrow M_2(\mathbb C),\qquad
T(z)=\frac1{\sqrt2}
\begin{pmatrix}
z_1+iz_2&z_3+iz_4\\
-z_3+iz_4&z_1-iz_2
\end{pmatrix}
\tag{AN.4}
\]
is unitary for the standard Hermitian and Hilbert–Schmidt forms. Its projectivization identifies \(Q^2\) with the nonzero rank-one matrices modulo scalar. The Segre map
\[
\begin{aligned}
\sigma:\mathbb {CP}^1\times\mathbb {CP}^1
&\longrightarrow\mathbb P(M_2(\mathbb C)),\\
([u],[v])&\longmapsto[uv^T]
\end{aligned}
\tag{AN.5}
\]
therefore gives a biholomorphism \(F=\mathbb P(T)^{-1}\circ\sigma\) onto \(Q^2\). At ambient scale \(\alpha\) it satisfies
\[
F^*g_{Q^2}=\alpha g_{\mathrm{FS}}\oplus\alpha g_{\mathrm{FS}}.
\tag{AN.6}
\]
Each projective-line factor has Gaussian curvature \(4/\alpha\); planes tangent to different factors have sectional curvature zero.

**Proof.** Expanding the four absolute squares of the entries of (AN.4), the mixed terms in each pair cancel, yielding
\[
\|T(z)\|_{\mathrm{HS}}^2=\sum_{j=1}^4|z_j|^2.
\]
The polarization identity proves preservation of the Hermitian form: its real part is obtained by applying this squared-norm equality to \(z+w,z-w\), and its imaginary part by applying it to \(z+iw,z-iw\). The inverse linear equations are
\[
\begin{aligned}
z_1&=(M_{11}+M_{22})/\sqrt2,&
z_2&=(M_{11}-M_{22})/(i\sqrt2),\\
z_3&=(M_{12}-M_{21})/\sqrt2,&
z_4&=(M_{12}+M_{21})/(i\sqrt2).
\end{aligned}
\tag{AN.7}
\]
Thus \(T\) is a unitary isomorphism. Direct multiplication gives
\[
\det T(z)=\frac12(z_1^2+z_2^2+z_3^2+z_4^2).
\tag{AN.8}
\]
A nonzero \(2\)-by-\(2\) matrix has determinant zero exactly when it has rank one: a nonzero column spans every column, by the vanishing determinant, and conversely dependent columns have zero determinant. This proves the image assertion for \(Q^2\).

The matrix \(uv^T\) is nonzero when \(u,v\ne0\); rescaling either vector merely rescales the matrix, so (AN.5) is well defined and holomorphic in projective charts. Every rank-one matrix has this form: choose a nonzero column \(u\), and write each column as a multiple of it, with those multiples comprising \(v^T\). The two projective vectors are uniquely determined. Indeed the image column line determines \([u]\), and then any nonzero row determines \([v]\). The inverse is holomorphic, not only set-theoretic: on a chart \(M_{ij}\ne0\), use the \(j\)-th column for \(u\) and set \(v_\ell=M_{i\ell}/M_{ij}\). These formulas give \(M=uv^T\) and have nonvanishing denominators on the stated chart. Their projective values agree on overlaps by uniqueness. This proves the global biholomorphism.

The projective transformation induced by \(T\) preserves \(g_{\mathrm{FS}}\), by its unitary invariance proved in L.1 or by (AM.7). It therefore remains to compute the pullback under \(\sigma\). On any product of standard projective charts choose holomorphic vector lifts \(u,v\) with one coordinate of each equal to one. The elementary identity
\[
\|uv^T\|_{\mathrm{HS}}^2
=\left(\sum_i|u_i|^2\right)\left(\sum_j|v_j|^2\right)
\tag{AN.9}
\]
and the pullback potential formula (AM.8) give
\[
\alpha\log\|uv^T\|_{\mathrm{HS}}^2
=\alpha\log\|u\|^2+\alpha\log\|v\|^2.
\]
The first term depends only on the first factor and the second only on the second. Their mixed Hessian has zero cross blocks and has precisely \(\alpha h_{\mathrm{FS}}\) on each diagonal block, by AL.1. This proves (AN.6) on a covering family of charts and hence globally.

Each factor has curvature \(4/\alpha\) by L.4. The Koszul formula in [Symmetric spaces A.1](symmetric-spaces.md#theorem-a-1) shows that the product metric has the product connection: vector fields lifted from different factors commute, have zero cross inner products, and their within-factor inner products are constant in the other factor's direction. Thus all cross covariant derivatives vanish and the connection in one factor is its own Levi-Civita connection. It follows that mixed curvature is zero. This gives the last assertion and proves, at the specified metric scale, the reducibility of \(Q^2\). □

## AO. Complex totally geodesic subspaces

**Theorem AO.1 (The complex tangent criterion).** Let \(f:N\to M\) be a connected totally geodesic immersed submanifold of a Hermitian symmetric space. Give \(N\) its induced Riemannian metric. Its local immersed branches are complex submanifolds if and only if
\[
J_{f(x)}\,df_x(T_xN)=df_x(T_xN)
\tag{AO.1}
\]
at one point \(x\in N\). In that case (AO.1) holds at every point, and the branches give \(N\) a complex atlas for which \(f\) is a holomorphic immersion. Completeness is not required for this criterion. In particular, a Lie triple system \(V\subseteq T_oM\) has a complete complex totally geodesic immersed image exactly when \(J_oV=V\).

**Proof.** The complex structure \(J\) of the ambient space is parallel by A.1–A.2. Total geodesy is equivalent to preservation of the tangent spaces by ambient parallel transport, by [Submanifolds G.1](submanifolds-and-hypersurfaces.md#proposition-g-1), since the Levi-Civita connection has zero torsion. If \(P\) is transport along any curve in \(N\), then \(PJ=JP\) and \(P\) identifies its endpoint tangent spaces. Thus invariance at one endpoint implies invariance at the other. A connected manifold is joined by finite geodesic polygons, by Hopf–Rinow E.1. This proves (AO.1) everywhere.

We justify complex coordinates on each branch explicitly. The homogeneous exponential charts of [Homogeneous spaces G.3](homogeneous-spaces-and-invariant-connections.md#lemma-g-3) are real analytic, and the invariant complex charts constructed in [Homogeneous spaces G.4](homogeneous-spaces-and-invariant-connections.md#theorem-g-4) are compatible with that real-analytic atlas. These apply to the Hermitian symmetric presentation of A.3; [Symmetric spaces B.3](symmetric-spaces.md#theorem-b-3) supplies such a presentation for a given Hermitian symmetric space, by its transvections, which preserve \(J\). In a symmetric presentation, [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) identifies the homogeneous curve \(\exp(tX)H\) with the Riemannian geodesic of initial vector \(X\). The local uniqueness argument in [Symmetric spaces C.3](symmetric-spaces.md#theorem-c-3) consequently identifies the branch through \(f(x)\) with
\[
v\longmapsto\exp_{f(x)}(v),\qquad
v\in df_x(T_xN)\text{ near }0.
\tag{AO.2}
\]
It is an analytic embedding in the homogeneous charts and hence in the compatible complex charts. Thus the branch is real analytic, rather than merely smooth.

Write its real dimension as \(2k\), which is even by (AO.1). In an ambient complex chart at the point, choose a complex-linear projection to \(\mathbb C^k\) that is an isomorphism on its complex tangent space. Such a projection is obtained by extending a complex basis of that tangent space to one of the ambient vector space. The analytic inverse theorem of [Curvature and holonomy I.1](curvature-and-holonomy-groups.md#lemma-i-1) makes the branch, locally, an analytic graph
\[
w\longmapsto (w,F(w)),\qquad w\in\mathbb C^k.
\]
Its tangent vectors have the form \((u,dF_wu)\). Invariance under multiplication by \(i\) says that \((iu,i\,dF_wu)\) is again a tangent vector; comparison of its first component forces
\[
dF_w(iu)=i\,dF_wu.
\tag{AO.3}
\]
The analytic Cauchy–Riemann proof in [Homogeneous spaces G.1](homogeneous-spaces-and-invariant-connections.md#lemma-g-1) now makes \(F\) holomorphic. These graph coordinates are therefore genuine complex submanifold coordinates. The change between two branch coordinates is a restriction of a holomorphic ambient coordinate change followed by a complex-linear projection, and is holomorphic; its inverse is obtained in the same way. Pulling the coordinates back to the local embedded branches of the immersion gives the claimed atlas on \(N\).

Conversely the derivative of a holomorphic immersion is complex linear, so its image tangent space is invariant under \(J\). This proves necessity as well. For a Lie triple system, [Symmetric spaces C.4](symmetric-spaces.md#theorem-c-4) supplies the connected complete immersed image, with its intrinsic orbit topology; its induced connection is the Levi-Civita connection by the metric restriction and zero torsion. Applying the criterion just proved gives the last assertion. For a zero-dimensional connected \(N\), the space is a point and the same conclusion holds with \(\mathbb C^0\). □

**Theorem AO.2 (All complete complex subspaces in rank one).** In projective space \(\mathbb {CP}^m\) with any positive multiple of the metric of L.1, the images of connected complete complex totally geodesic immersions are exactly its projective linear subspaces \(\mathbb P(E)\), where \(E\subseteq\mathbb C^{m+1}\) is a nonzero complex linear subspace. Each such immersion is an embedding onto its image.

In the complex hyperbolic ball \(\mathcal D_{m,1}\) of L.2, with any positive multiple of its metric, the corresponding images are exactly the holomorphic isometric translates of
\[
\{(z_1,\ldots,z_k,0,\ldots,0):
 |z_1|^2+\cdots+|z_k|^2<1\},
\qquad 0\leq k\leq m.
\tag{AO.4}
\]
Again each complete immersion is an embedding onto that image. The case \(k=0\) means a point. In the positive-line interpretation of AK.1, these are the positive lines in complex linear subspaces of signature \((1,k)\).

**Proof.** First fix a point \(o\) and a complex \(k\)-dimensional tangent subspace \(V\). By the transitive holomorphic isometries of L.1–L.2 we may put \(o\) at the reference line or at the origin of the ball. Its tangent space is then \(\mathbb C^m\). A unitary transformation sends \(V\) to its first \(k\) coordinates: choose an orthonormal basis of \(V\), extend it by orthonormal vectors in its perpendicular complement, and use the resulting unitary matrix. This is an allowed stabilizer action, by L.1–L.2. Therefore it suffices to construct and identify the coordinate subspace with that tangent.

For projective space let
\[
E=\operatorname{span}_{\mathbb C}(e_0,e_1,\ldots,e_k).
\]
The inclusion \(\mathbb P(E)\subseteq\mathbb {CP}^m\) is holomorphic and embedded in the usual projective charts. Its induced metric is the same scaled projective metric in dimension \(k\): insert a graph vector with its last \(m-k\) entries zero in (L.3). The inverse matrices restrict to the corresponding first blocks and the zero blocks contribute nothing to the trace. It is complete by L.1.

It is totally geodesic as follows. The unitary transformation equal to \(+I\) on \(E\) and \(-I\) on \(E^\perp\) induces an ambient isometry fixing \(\mathbb P(E)\) pointwise and acting as the identity on its tangent vectors. Its projective fixed points are exactly \(\mathbb P(E)\) and, when \(E^\perp\ne0\), \(\mathbb P(E^\perp)\). Indeed a vector with nonzero components in both eigenspaces cannot be multiplied by a single scalar to equal its image. An ambient geodesic initially tangent to \(\mathbb P(E)\) is fixed by the isometry, by uniqueness of the geodesic equation, [Geodesics A.1](geodesics-normal-coordinates-and-curvature.md#theorem-a-1). Locally at its starting point it therefore lies in \(\mathbb P(E)\), because the other fixed subspace is closed and disjoint from that point. This proves total geodesy; when \(E\) is the whole space it is immediate.

The tangent-generated algebra is also explicit. For \(k\geq1\), its compact odd matrices occupy only the \((k+1)\)-coordinate block \(E\); their even brackets and triple brackets stay in that block. AK.1, applied with \(p=1,q=k\), proves that they generate exactly \(\mathfrak{su}(k+1)\): the odd entries, their even off-diagonal brackets and diagonal differences span that algebra. The corresponding block subgroup has projective orbit \(\mathbb P(E)\), by the same transitivity proof. The noncompact block is \(\mathfrak{su}(1,k)\), with its opposite odd bracket proved in L.3 and its full orbit given by AK.1. For \(k=0\), the tangent-generated algebra is zero and the orbit is a point.

For the ball, the linear unitary isometry that keeps the first \(k\) coordinates and negates the others has fixed set exactly (AO.4). The same geodesic uniqueness proves total geodesy. Restriction of (L.12) to those coordinates gives exactly the dimension-\(k\) ball metric, which is complete by L.2. The inclusion is holomorphic and embedded. In both settings this constructs a complete subspace \(P\) through \(o\) with tangent \(V\); the limiting cases \(k=0,m\) are included.

Now let \(f:N\to M\) be a connected complete totally geodesic immersion with a complex tangent space at \(x\), and put \(o=f(x)\), \(V=df_x(T_xN)\). AO.1 makes it a complex immersion. For every \(v\in T_xN\), total geodesy and the geodesic uniqueness theorem give
\[
f(\exp_x^N v)=\exp_o^M(df_xv).
\tag{AO.5}
\]
Both sides are defined at every \(v\). By Hopf–Rinow B.2, the exponential at \(x\) is onto \(N\), so
\[
f(N)=\exp_o^M(V).
\tag{AO.6}
\]
Apply exactly the same reasoning to the complete coordinate subspace \(P\) constructed above. Its exponential is onto and its geodesics are ambient geodesics, hence \(P=\exp_o^M(V)\). Thus \(f(N)=P\), proving the claimed classification of images.

There is no hidden multiple covering in this statement. Regarding \(f\) as a map \(N\to P\), its derivative is injective and the two dimensions agree, so it is a local diffeomorphism. The induced metric makes it a local isometry, hence affine by the Koszul uniqueness in [Symmetric spaces A.1](symmetric-spaces.md#theorem-a-1). Hopf–Rinow E.2 makes it a covering. The projective subspace is simply connected by AK.1, and the ball is simply connected by the convex contraction in L.2; the zero-dimensional case is a point. [Flat connections D.3](flat-connections-and-infinitesimal-holonomy.md#theorem-d-3) says that a connected covering of a simply connected space is a diffeomorphism. This proves that \(f\), as a map into the ambient manifold, is an embedding onto \(P\).

Finally, in the positive-line model, the coordinate subball consists of positive lines in \(\mathbb C e_0\oplus\mathbb C^k\), whose restricted form has signature \((1,k)\). An isometry preserves the ambient Hermitian form, so its translates have the asserted description. Conversely in any such subspace choose a positive unit vector. By the transitivity in L.2, move its line to the first coordinate line; a unit scalar adjusts the chosen vector itself. Its ambient orthogonal complement is now the negative definite last \(m\) coordinates. Removing the first component of vectors in the given subspace consequently leaves a negative definite \(k\)-space there. Gram–Schmidt for the negative of the restricted form gives an orthonormal negative basis. Extend it within the ambient negative perpendicular complement. The resulting form-preserving basis change sends the subspace to the coordinate model and acts by the holomorphic isometries of L.2. This proves the final equivalence. □

## AP. The compact classical quotient models

Use the compact quaternionic group
\(\mathrm{Sp}(n)=\{Q\in M_n(\mathbb H):Q^\dagger Q=I\}\).
Its complex realization is (AC.7). Write
\[
J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix},
\qquad
S_{p,q}=\operatorname{diag}(I_p,-I_q).
\tag{AP.1}
\]
Conjugation by a displayed matrix is denoted \(\operatorname{Ad}\). In the CI row below, \(iI_n\) is a quaternionic matrix. In the DIII row, \(J_n\) is real.

**Theorem AP.1 (Actual quotients and their covering interpretation).** The seven compact classical families in AE.2 have the following connected compact homogeneous models, with the indicated group involutions:

| Family | Group \(G\) | Connected subgroup \(H\) | Involution \(\sigma\) on \(G\) |
| --- | --- | --- | --- |
| AI, \(n\geq2\) | \(SU(n)\) | \(SO(n)\) | \(g\mapsto\bar g\) |
| AII, \(n\geq2\) | \(SU(2n)\) | \(\mathrm{Sp}(n)\) in (AC.7) | \(g\mapsto J_n\bar gJ_n^{-1}\) |
| AIII, \(p,q\geq1\) | \(SU(p+q)\) | \(S(U(p)\times U(q))\) | \(\operatorname{Ad}S_{p,q}\) |
| BDI, \(p,q\geq1\) | \(SO(p+q)\) | \(SO(p)\times SO(q)\) | \(\operatorname{Ad}S_{p,q}\) |
| CI, \(n\geq1\) | \(\mathrm{Sp}(n)\) | \(U(n)\), with complex entries | \(\operatorname{Ad}(iI_n)\) |
| CII, \(p,q\geq1\) | \(\mathrm{Sp}(p+q)\) | \(\mathrm{Sp}(p)\times\mathrm{Sp}(q)\) | \(\operatorname{Ad}S_{p,q}\) |
| DIII, \(n\geq3\) | \(SO(2n)\) | \(U(n)\), acting on \(\mathbb R^{2n}\) | \(\operatorname{Ad}J_n\) |

In every row, \(H=(G^\sigma)^0\). In all rows except BDI it is the full fixed group. These quotients are smooth complete Riemannian symmetric spaces and realize exactly the compact algebraic pairs and dimensions of AE.2. Their simply connected Riemannian covers are the corresponding spaces in AJ.1–AJ.2. Thus the table specifies actual models and a precise covering operation, without assuming that an arbitrary stabilizer with the same Lie algebra gives the same quotient.

The AIII and BDI models are respectively the complex Grassmannians of AK.1 and the oriented real Grassmannians of AK.2. CI and DIII carry invariant complex structures for their full displayed stabilizers, are Hermitian symmetric and are themselves simply connected. On each irreducible row, choosing a positive multiple of the tangent metric gives every invariant metric scale in the classification. BDI at \((1,1)\) is the flat circle with simply connected cover a line; BDI at \((2,2)\) has the two rank-one factors already identified in AN.2.

**Proof.** [De Rham decomposition Y.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-y-2) proves that all displayed acting groups are compact connected embedded matrix Lie groups, and proves connectedness of \(SO(n),U(n),SU(n)\) and \(\mathrm{Sp}(n)\) in every size used here. [De Rham decomposition Y.3](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-y-3) and AC.2 identify the compact complex symplectic realization with \(\mathrm{Sp}(n)\) and prove its inclusion in \(SU(2n)\). Products of the connected subgroups are connected by taking their paths componentwise. The AIII stabilizer's connectedness was separately proved in AK.1, including its determinant constraint.

We check the fixed subgroups at group level. For AI, a real matrix in \(SU(n)\) is exactly a real orthogonal matrix of determinant one. For AII, the fixed equation is
\[
gJ_n=J_n\bar g.
\tag{AP.2}
\]
By AC.2 its solutions are precisely matrices of the form (AC.7); unitarity is exactly \(Q^\dagger Q=I\). Conversely all such matrices have complex determinant one by [De Rham decomposition Y.3](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-y-3), so there is no additional determinant restriction. This gives the full \(\mathrm{Sp}(n)\).

For AIII, commuting with \(S_{p,q}\) forces the off-diagonal blocks to vanish; unitarity and determinant one give exactly \(S(U(p)\times U(q))\). For BDI the same computation gives the full fixed group
\[
S(O(p)\times O(q))
=\{(a,d):\det a\,\det d=1\}.
\tag{AP.3}
\]
The signs of the two determinants cannot change along a path. Its component containing the identity is exactly \(SO(p)\times SO(q)\), since that product is connected. Thus BDI uses the connected component, a genuine distinction from the other rows.

For CI, an entry of a quaternionic matrix commutes with \(i\) exactly when it is complex. Indeed, in \(a+b\mathbf j\), the relation \(\mathbf j i=-i\mathbf j\) makes commutation equivalent to \(b=0\). The fixed unitary matrices are consequently exactly \(U(n)\). For CII, the eigenspaces of \(S_{p,q}\) force block diagonal quaternionic matrices, with independent compact symplectic blocks.

For DIII, commuting with \(J_n\) means being complex linear when \(J_n\) is the complex structure on \(\mathbb R^{2n}\). A real orthogonal map commuting with it preserves the associated Hermitian form: both the real scalar product and the form \((v,w)\mapsto\langle J_nv,w\rangle\) are preserved. Conversely a unitary map preserves that real scalar product. Such a map has real determinant \(+1\), since \(U(n)\) is connected and contains the identity, by [De Rham decomposition Y.2](holonomy-and-the-de-rham-decomposition-theorem.md#theorem-y-2). Therefore intersection with \(SO(2n)\) removes no element and the full fixed group is \(U(n)\).

All displayed automorphisms square to the identity. This is immediate for conjugation and the sign matrices. In AII, \(J_n\) is real and \(J_n^2=-I\), so applying the formula twice gives conjugation by \(-I\), hence the identity. In CI and DIII the conjugating matrix squares to the central scalar \(-I\), with the same consequence. The formulas preserve each group by their defining unitary, orthogonal or quaternionic equations. Their fixed groups and connected components are closed, and hence the displayed \(H\)'s are closed subgroups. [Symmetric spaces B.1](symmetric-spaces.md#lemma-b-1) now supplies the smooth quotients and their tangent eigenspaces.

Differentiating the displayed group formulas gives exactly the algebra involutions in AD.1–AD.3. Those complete calculations identify each fixed algebra, odd space, real dual and dimension; no pair is inferred from its label. To construct a positive invariant metric explicitly, use the faithful complex representation \(\rho\) of the acting group, with real matrices regarded as complex and quaternionic matrices represented by (AC.7). On its odd algebra put
\[
b(X,Y)=-\operatorname{Re}\operatorname{tr}
                  \bigl(\rho(X)\rho(Y)\bigr).
\tag{AP.4}
\]
Its matrices are skew-Hermitian, so
\[
b(X,X)=\operatorname{tr}(\rho(X)^*\rho(X))>0
\quad\text{for }X\ne0.
\]
The last trace is the sum of squared absolute values of the entries, and faithfulness makes it strictly positive. Cyclicity of trace proves invariance under the entire \(H\). [Symmetric spaces B.2](symmetric-spaces.md#theorem-b-2) gives its complete Levi-Civita connection, global point symmetries and curvature \(-[[X,Y],Z]\). Compactness and connectedness follow from those of \(G\) and continuity and surjectivity of the quotient map. AIII and BDI are the already proved actual identifications AK.1–AK.2.

For CI the compact odd matrices in (AC.7) have the form
\[
X_B=\begin{pmatrix}0&B\\-\bar B&0\end{pmatrix},
\qquad B^T=B,
\tag{AP.5}
\]
and the full stabilizer is \(\operatorname{diag}(U,\bar U)\).
Its action sends \(B\) to \(UBU^T\); thus \(B\mapsto iB\) commutes with the whole stabilizer, squares to \(-I\) and preserves (AP.4). A.3 gives the invariant complex coordinates and complete Hermitian symmetric structure.

For DIII the odd real matrices satisfy \(XJ_n=-J_nX\). The map \(X\mapsto J_nX\) preserves that space and its skew symmetry:
\[
(J_nX)^T=XJ_n=-J_nX.
\]
It squares to \(-I\). It commutes with conjugation by every element of \(H\), since these commute with \(J_n\). Orthogonality of \(J_n\), or direct substitution in the trace form, shows that it preserves (AP.4). A.3 again applies. The effective algebras in these ranges are compact semisimple by AC.1 and AE.2. Hence I.2 proves simple connectivity of these actual Hermitian quotients. The same property for AIII was proved in AK.1.

For the covering assertion in every row, take the smooth universal cover supplied by [Flat connections D.2](flat-connections-and-infinitesimal-holonomy.md#theorem-d-2) and pull back the metric. The covering is a local isometry, so its pulled-back connection is Levi-Civita by [Symmetric spaces A.1](symmetric-spaces.md#theorem-a-1). Hopf–Rinow E.3 lifts geodesic completeness from the quotient to the cover. Curvature remains parallel and has the same value on the identified tangent space. It is therefore exactly the simply connected model of that metric pair by AJ.1, or equivalently by the metric-pair uniqueness used in AJ.2. This proves the precise global interpretation of the table. In each irreducible nonflat case, D.1 identifies every invariant positive form with a positive multiple of the Killing form; (AP.4) supplies one such form. Its positive multiples give all scales.

Finally for BDI \((1,1)\), \(G=SO(2)\) and \(H\) is trivial. Its odd algebra is abelian, so the connection is flat. The circle and its line cover were explicitly treated in G.1. For \((2,2)\), AM.1 identifies the quotient with the quadric surface and AN.2 gives its global product at the specified metric normalization; its underlying algebraic splitting is AE.2. This also records the boundary cases that cannot be read as irreducible nonflat rows. □

## AQ. The compact Siegel dual at the transferred scale

**Theorem AQ.1 (The compact CI space and the normalized dual).** The compact quotient \(\mathrm{Sp}(r)/U(r)\), \(r\geq1\), of AP.1 is the compact Hermitian symmetric dual of the actual Siegel space \(\mathcal H_r\) of M.1–M.3. In the compact odd coordinates \(X_B\) of (AP.5), its complex structure and the metric transferred from (M.10) are
\[
JB=iB,\qquad
b_c(X_B,X_D)=4\operatorname{Re}\operatorname{tr}(B^*D),
\qquad B^T=B,\quad D^T=D.
\tag{AQ.1}
\]
Both spaces are simply connected. The identification of the noncompact dual preserves the full tangent metric and complex structure, not only the real Lie algebra. For \(r=1\), the compact space is a projective line with metric \(4g_{\mathrm{FS}}\) and Gaussian curvature \(1\); its dual is the curvature-\(-1\) disc or upper half-plane of L.4.

**Proof.** The full compact stabilizer and its action \(B\mapsto UBU^T\) were proved in AP.1. This action is complex linear and preserves the Hermitian trace form: unitarity gives \(\bar U U^T=I\), and cyclicity then gives
\[
\operatorname{tr}((UBU^T)^*(UDU^T))
=\operatorname{tr}(B^*D).
\]
Thus (AQ.1) defines a positive full-stabilizer invariant Hermitian form. A.3 constructs its complex coordinates, parallel complex structure and complete symmetric metric; I.2, as in AP.1, proves simple connectivity.

We exhibit the real dual and its tangent identification. On the real symplectic group, the map \(g\mapsto(g^T)^{-1}\) is an involutive automorphism. In fact the equation \(g^T\Omega g=\Omega\) gives \((g^T)^{-1}=\Omega g\Omega^{-1}\); conjugation by the symplectic matrix \(\Omega\) preserves the group and squares to conjugation by \(-I\). Its fixed points are the orthogonal symplectic matrices. Orthogonality reduces the symplectic equation to commutation with \(\Omega\), which forces the blocks \(\begin{pmatrix}a&b\\-b&a\end{pmatrix}\); the remaining orthogonality equations are exactly (M.9). Thus its full fixed group is precisely the \(U(r)\) of M.4. Its differential is \(L\mapsto-L^T\).

Put
\[
C=\frac1{\sqrt2}
\begin{pmatrix}I&-iI\\ I&iI\end{pmatrix}.
\tag{AQ.2}
\]
Its conjugate transpose is its inverse, by multiplication. A general real symplectic Lie-algebra matrix is
\[
L=\begin{pmatrix}A&F\\G&-A^T\end{pmatrix},
\qquad F^T=F,\quad G^T=G,
\tag{AQ.3}
\]
as follows by differentiating \(g^T\Omega g=\Omega\) from M.1. Block multiplication gives
\[
\begin{aligned}
CLC^{-1}&=\begin{pmatrix}P&Q\\\bar Q&\bar P\end{pmatrix},\\
P&=\frac{A-A^T+i(F-G)}2,\\
Q&=\frac{A+A^T-i(F+G)}2.
\end{aligned}
\tag{AQ.4}
\]
Here \(P^*=-P\) and \(Q^T=Q\). Conversely such \(P,Q\) recover real matrices in (AQ.3): the real and imaginary parts determine \(A-A^T,F-G,A+A^T,F+G\), and addition and subtraction give \(A,F,G\). Thus (AQ.4) identifies the entire real symplectic algebra with these blocks.

The compact symplectic algebra is
\[
\left\{
\begin{pmatrix}P&B\\-\bar B&\bar P\end{pmatrix}:
P^*=-P,\ B^T=B\right\},
\tag{AQ.5}
\]
by AC.2. Its even blocks are exactly the \(P\)-blocks in (AQ.4). Replace its odd matrix \(X_B\) by
\[
Y_B=\begin{pmatrix}0&B\\\bar B&0\end{pmatrix}.
\]
The even-odd bracket sends \(B\) to \(PB-B\bar P\) on both sides. Multiplication of two odd blocks shows that
\[
[Y_B,Y_D]=-[X_B,X_D].
\tag{AQ.6}
\]
The even-even bracket is unchanged. This proves the precise opposite-odd-bracket duality of D.3, with the same odd coordinate \(B\).

In the real coordinates (AQ.3), an odd matrix is
\[
L_{X,Y}=\begin{pmatrix}X&Y\\Y&-X\end{pmatrix},
\qquad X^T=X,\quad Y^T=Y.
\tag{AQ.7}
\]
Formula (AQ.4) identifies its odd coordinate as \(B=X-iY\). Differentiate the fractional action of M.1 at \(W=iI\). Its tangent is
\[
\delta W=2(Y+iX)=2iB.
\tag{AQ.8}
\]
Indeed the derivative of \((aW+b)(cW+d)^{-1}\) at the identity is \(a'W+b'-W(c'W+d')\); substitution from (AQ.7) yields (AQ.8).

At \(W=iI\), formula (M.10) is the ordinary Hermitian trace form on symmetric tangent matrices. Substitution of \(\delta W=2iB\) and \(2iD\) therefore gives exactly \(4\operatorname{tr}(B^*D)\), whose real part is (AQ.1). The derivative at \(iI\) of the Cayley coordinate \(Z=(W-iI)(W+iI)^{-1}\) is \(\delta Z=\delta W/(2i)=B\). Thus multiplication by \(i\) in the bounded holomorphic coordinate is exactly \(B\mapsto iB\), including its sign. This proves both the complex and metric normalization.

For the full stabilizer, a real matrix of M.4 is
\(\begin{pmatrix}a&b\\-b&a\end{pmatrix}\).
Conjugating it by (AQ.2) gives
\(\operatorname{diag}(a+ib,a-ib)\), and \(a+ib\) ranges over all of \(U(r)\). Its action on odd coordinates is therefore exactly \(B\mapsto UBU^T\), the compact stabilizer action. M.1 proves that the actual real group is connected and that this is its full stabilizer; M.2 proves the complete invariant Siegel metric. The domain is convex, hence simply connected by M.1. Its effective pair is precisely (AQ.4), so the existence and uniqueness theorem AJ.1 identifies it with the simply connected noncompact dual, with the metric and complex structure just checked. No covering ambiguity remains.

When \(r=1\), (AC.7) identifies \(\mathrm{Sp}(1)\) with \(SU(2)\): its matrices have the form
\(\begin{pmatrix}a&b\\-\bar b&\bar a\end{pmatrix}\),
with \(|a|^2+|b|^2=1\), and every determinant-one unitary \(2\)-by-\(2\) matrix has this form by its orthonormal columns. The stabilizer of the second coordinate line is exactly the diagonal \(U(1)\). Its orbit map therefore gives the projective line, by AK.1. The derivative of \(X_B\) on that line is the affine graph coordinate \(B\), so it preserves the complex structure in (AQ.1). At the origin its metric is \(4|dB|^2\), hence by transitivity it is \(4g_{\mathrm{FS}}\) everywhere. L.4 gives its curvature \(1\) and the corresponding dual curvature \(-1\).

More generally, the AI \(n=2\), AIII \(p=q=1\), and CI \(r=1\) algebraic coincidences proved in AE.2 identify their simply connected metric realizations by AJ.1, after the positive scale is fixed. The projective-line and disc models above supply those realizations; their round-sphere interpretation is the space-form classification in [Sectional curvature D.2](sectional-curvature-and-space-forms.md#theorem-d-2). Thus the small-rank repetitions in the classical table are global metric statements with specified scales. □

## Further reading

- J.-H. Eschenburg, *Lecture Notes on Symmetric Spaces*, [freely accessible author-hosted notes](https://myweb.rz.uni-augsburg.de/~eschenbu/symspace.pdf), §1 Examples 6–8 and §§4–8, on Grassmannians, classical quotient models, totally geodesic subspaces, symmetric presentations, Lie triple systems, simply connected realizations, Killing forms, curvature signs and duality.

- Eckhard Meinrenken, *Lie groups and Lie algebras (Winter 2026)*, [freely accessible University of Toronto lecture notes](https://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf), §8.5 and §9.2, on bi-invariant metrics, maximal tori and compact-group structure.

- A. Iozzi and Y. Krifka, *Symmetric Spaces*, Spring 2021, [Solution Exercise Sheet 5](https://metaphor.ethz.ch/x/2021/fs/401-3226-00L/ex/sol5.pdf), Exercise 3, on the symplectic action and the Siegel upper half-space.
- Ngaiming Mok, *Complex Geometry on Bounded Symmetric Domains, I. Metric rigidity*, 21 August 2006, [author-hosted lecture slides](https://hkumath.hku.hk/~nmok/I.pdf), page 2, for the rectangular matrix-ball model.

- Pavel Etingof, *Lie Groups and Lie Algebras*, MIT 18.755, Spring 2024, [freely accessible MIT OpenCourseWare notes](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf), §§6.2–6.3, §11.4, §§13.1–13.2, §14.2, §§19.1–19.4, §§20.1–20.3, §21.4, §§22.1–22.2, §§23.1–23.2, §23.8, §§24.1–24.2, §40.1, §41.1, §§41.3–41.4 and §§42.1–42.2, on rank-one representations, ordered words, free Lie algebras, Cartan subalgebras, roots, finite diagrams, the Serre presentation, compact real forms, classical matrix algebras, involutions, exceptional fixed algebras and the E6 diagram-reversal cases.

- Sebastian Klein, *The complex quadric from the standpoint of Riemannian geometry*, doctoral thesis, University of Cologne, 2005, [open-access institutional version](https://kups.ub.uni-koeln.de/1627/1/Diss.pdf), §§1.1–1.2, §3.2, §3.4 and §8.1, on the quadric, its oriented-plane description and the low-dimensional models.

- Ravi Vakil, *Foundations of Algebraic Geometry*, Classes 15 and 16, 12 November 2007, updated 10 December 2007, [author-hosted notes](https://math.stanford.edu/~vakil/0708-216/216class1516.pdf), §5, especially Example 5.1, on the Segre map and the quadric surface.

The preceding lessons linked in the proofs supply the analytic, representation-theoretic and geometric foundations used here.