Classification of real forms by Vogan diagrams

For a real form choose $\theta$ by [L1]. To ensure an initial theta-stable Cartan exists, choose a maximal abelian subspace $\mathfrak{t}\subseteq {\mathfrak{k}}_0$ and then a maximal abelian subspace $\mathfrak{a}\subseteq Z_{{\mathfrak{p}}_0}(\mathfrak{t})$, maximizing finite dimensions. Put ${\mathfrak{h}}_0=\mathfrak{t}\oplus \mathfrak{a}$. Its real centralizer is itself: a centralizing compact component lies in $\mathfrak{t}$ by maximality, and the split component lies in $\mathfrak{a}$. Invariance gives $(\mathrm{ad}X{)}^{\ast }=-\mathrm{ad}(\theta X)$ for $B_{\theta }$, so the commuting adjoints of ${\mathfrak{h}}_0$ are normal and simultaneously diagonalizable. The zero simultaneous space in the complexification is $\mathfrak{h}={\mathfrak{h}}_{0\mathbb{C}}$. If $Z$ normalizes $\mathfrak{h}$, its nonzero weight component $Z_{\lambda }$ satisfies $[H,Z_{\lambda }]=\lambda (H)Z_{\lambda }$; normalization forces it to vanish for some $H$ with $\lambda (H)\mathrm{\ne }0$. Thus the normalizer is $\mathfrak{h}$. This proves that ${\mathfrak{h}}_0$ is a theta-stable Cartan. Apply the finite Cayley process in [L2] to obtain a maximally compact one. It has no real roots, so every root restricts nontrivially to $i{\mathfrak{t}}_0$. Choose a point there off the finitely many root kernels. Positivity at that point defines a theta-stable compatible base. These choices exist under [A1].

Conversely take an abstract diagram with permutation $\tau$ and set ${\epsilon }_i=-1$ on painted fixed vertices, $+1$ on all other vertices. Use the compact conjugation and simple triples of [L5]. The assignment $\mathrm{\Theta }{e}_i={\epsilon }_i{e}_{\tau i}$, $\mathrm{\Theta }{f}_i={\epsilon }_i{f}_{\tau i}$, $\mathrm{\Theta }{h}_i=h_{\tau i}$ preserves every Serre relation: $\tau$ preserves the Cartan matrix; the mixed bracket has coefficient ${\epsilon }_i^2=1$ when $i=j$; and each positive or negative Serre relation maps to a nonzero scalar multiple of the corresponding permuted relation. By [L4] it extends to an endomorphism. Its square fixes every $e_i,f_i,h_i$ because ${\epsilon }_i{\epsilon }_{\tau i}=1$, so it is an involutive automorphism. Moreover $\mathrm{\Theta }\kappa =\kappa \mathrm{\Theta }$ on all these generators, hence on $\mathfrak{g}$. Thus it preserves the compact form $\mathfrak{u}$.

By [L1] the diagram class is independent of the Cartan and positive system for fixed $\theta$. Conjugating one Cartan involution to another transports all this data. A real Lie isomorphism likewise transports the Killing form, involution, root spaces and painting. Hence the assignment is well defined on real isomorphism classes.

Fix such a triple and write $\sigma$ for conjugation of its real form. The involutions $\sigma ,\theta$ commute, and $\kappa =\theta \sigma$ has fixed algebra $\mathfrak{u}={\mathfrak{k}}_0\oplus i{\mathfrak{p}}_0$. The bracket inclusions show closure, and [L2] shows that $B$ is negative definite there. Thus $\mathfrak{u}$ is a compact real form, with positive Hermitian form $-B(Z,\kappa Z)$. On $\mathfrak{h}$, $\kappa$ negates the real span of the coroots and carries the $\alpha$ root space to the $-\alpha$ root space. For each simple root choose nonzero $e_i$, set $f_i=-\kappa e_i$, and rescale both by the same positive number until $B(e_i,f_i)=2\mathrm{/}({\alpha }_i,{\alpha }_i)$. Positivity makes this possible; [L3] then gives $[e_i,f_i]=h_i$ and $\kappa f_i=-e_i$.

Let ${\mathfrak{u}}^{\pm }$ be its real eigenspaces. The antilinear involution $\mathrm{\Sigma }=\mathrm{\Theta }\kappa$ has fixed algebra ${\mathfrak{g}}_D={\mathfrak{u}}^{+}\oplus i{\mathfrak{u}}^{-}$. For any $Z$, the identity $Z=(Z+\mathrm{\Sigma }Z)\mathrm{/}2+i(Z-\mathrm{\Sigma }Z)\mathrm{/}(2i)$ proves $\mathfrak{g}={\mathfrak{g}}_D\oplus i{\mathfrak{g}}_D$; the intersection is zero because $\mathrm{\Sigma }$ is antilinear. Brackets are preserved, and [L7] gives semisimplicity. If $X=K+iP$ with $K\in {\mathfrak{u}}^{+}$, $P\in {\mathfrak{u}}^{-}$, orthogonality of the eigenspaces gives $-B(X,\mathrm{\Theta }X)=-B(K,K)-B(P,P)>0$ for $X\mathrm{\ne }0$. Thus $\mathrm{\Theta }$ is a Cartan involution. The stable Cartan $\mathfrak{h}$ is preserved by both $\mathrm{\Theta }$ and $\kappa$, so its fixed real part ${\mathfrak{h}}_D$ complexifies to $\mathfrak{h}$ by the same decomposition. It is abelian; a real normalizer complexifies into the normalizer $\mathfrak{h}$, so it is a real Cartan.

Write $\theta e_i=a_i{e}_{\tau i}$, where $\tau$ is the diagram involution. Commutation with $\kappa$ gives $\theta f_i=\overline{a_i}{f}_{\tau i}$. Since $\theta$ preserves $B$ and root lengths, the normalization in step 2.2 gives $\mathrm{\mid }{a}_i\mathrm{\mid }=1$; also $a_i{a}_{\tau i}=1$. At a fixed vertex $a_i={\epsilon }_i\in \{1,-1\}$ is precisely its painting sign. In each two-element orbit choose one index $i$ and replace $e_{\tau i}$ by $a_i{e}_{\tau i}$ and $f_{\tau i}$ by $a_i^{-1}{f}_{\tau i}$. The normalization and compact conjugation are unchanged because $\mathrm{\mid }{a}_i\mathrm{\mid }=1$. Now $\theta$ interchanges both positive generators and both negative generators in that pair with coefficient one. Its effect on every $h_i$ is $h_i\mapsto h_{\tau i}$.

The induced action on roots is $\tau$, which preserves the positive system. A real root would satisfy $\tau \alpha =-\alpha$, impossible after choosing its positive sign. Hence ${\mathfrak{h}}_D$ is maximally compact by [L2]. To check compatibility in the compact-first convention, take an interior point $H$ of the positive chamber in the real coroot span and replace it by $(H+\mathrm{\Theta }H)\mathrm{/}2$. Every positive root remains positive on this point, which belongs to the fixed subspace $i({\mathfrak{h}}_D\cap {\mathfrak{u}}^{+})$. Thus this base is obtainable by taking the compact directions first. On fixed simple root spaces, $\mathrm{\Theta }$ has exactly the specified signs; on paired simple spaces it interchanges them. This proves that the resulting diagram is exactly the given abstract diagram. There is no requirement that every fixed nonsimple root be generated by fixed simple roots.

Two real forms with isomorphic diagrams have, after relabeling, the same Cartan matrix, the same permutation $\tau$ and the same fixed-vertex signs. Use step 3.1 separately in the two forms. By [L4], the map taking each $e_i,f_i,h_i$ to its namesake is a complex Lie isomorphism $F$ (the inverse generator assignment proves invertibility). It intertwines $\theta$ and ${\theta }^{\mathrm{\prime }}$ and also the antilinear $\kappa$ and ${\kappa }^{\mathrm{\prime }}$: the identities hold on all positive and negative simple generators and on the Cartan generators, hence on their brackets and complex linear combinations. Therefore it intertwines $\sigma =\theta \kappa$ and ${\sigma }^{\mathrm{\prime }}={\theta }^{\mathrm{\prime }}{\kappa }^{\mathrm{\prime }}$, and restricts to an isomorphism of their real fixed algebras. No common embedding of their Cartans or compact forms was assumed.

Every allowed painted reflection is realized by changing the positive system of the same real form. Let $\alpha$ be a painted simple root. Since $\theta \alpha =\alpha$, $s_{\alpha }$ commutes with $\theta$, so $s_{\alpha }\mathrm{\Delta }$ is compatible. The new root $-\alpha$ is still painted, since $\theta f_{\alpha }=-f_{\alpha }$. For a different fixed simple root $\beta$ put $m=-\beta (h_{\alpha })\ge 0$. The string is $\beta ,\dots ,\beta +m\alpha$, with one-dimensional root spaces; its ${\mathfrak{s}\mathfrak{l}}_2$ weights are $-m,-m+2,\dots ,m$. By [L6] this is an irreducible string (two irreducible summands of that parity would have overlapping centered weight intervals). Thus $(\mathrm{ad}{e}_{\alpha }{)}^m{e}_{\beta }\mathrm{\ne }0$ spans the root space for $s_{\alpha }\beta$. Its theta sign is $(-1{)}^m{\epsilon }_{\beta }$. This is exactly the parity toggle of Vogan diagram. Two-element orbits remain two-element orbits because $s_{\alpha }$ commutes with $\theta$, and remain unpainted. Diagram isomorphisms merely relabel. Applying these observations along a finite equivalence chain, and then step 4.1, proves injectivity.

Steps 2.1 and 5.1 give a well-defined injection, and step 3.2 gives surjectivity. When $\mathfrak{g}=0$ the empty diagram and zero real form are unique and every construction uses empty generator sets. Rank one permits either painting sign, with no paired vertices or other vertices to toggle. Disconnected diagrams are covered by the same Serre argument, including permutations of isomorphic components. The claimed bijection follows under [A1].