Vogan diagrams for real forms of sl three c

On ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ the maps $X\mapsto -X^{\ast }$, $X\mapsto -I{X}^{\ast }I$ and $X\mapsto \bar{X}$ are conjugate-linear involutions preserving brackets: conjugate transpose reverses products and its extra minus sign restores the commutator. Their fixed algebras are respectively the three displayed algebras, so [L2] proves they are real forms and semisimple. On the first algebra $\theta =1$ is a Cartan involution since $-B(X,X)=6\mathrm{tr}(X{X}^{\ast })>0$ for nonzero skew-Hermitian $X$. On the second $\theta X=IXI=-X^{\ast }$, which is a real automorphism squaring to one and has $B_{\theta }(X,X)=6\mathrm{tr}(X{X}^{\ast })>0$. On the third $\theta X=-X^{\mathsf{T}}$ has $B_{\theta }(X,X)=6\mathrm{tr}(X{X}^{\mathsf{T}})>0$. The real diagonal traceless Cartan has real adjoint eigenvalues, so this third real form is split.

In both unitary forms take the diagonal traceless skew-Hermitian Cartan. Its complexification is the diagonal Cartan of [L1]; it is abelian and its real normalizer lies in the real part of the complex normalizer, hence equals itself. It is fixed pointwise by $\theta$, so every root is imaginary and there are no real roots; it is maximally compact by [L3]. For $\mathfrak{s}\mathfrak{u}(3)$ the complex extension of $\theta$ is the identity, so both simple roots are compact and neither is painted. For $\mathfrak{s}\mathfrak{u}(2,1)$ the complex extension is $X\mapsto IXI$, with $\theta E_{ij}=I_{ii}{I}_{jj}{E}_{ij}$. Thus $\mathbb{C}{E}_{12}$ is in $\mathfrak{k}$ and $\mathbb{C}{E}_{23}$ is in $\mathfrak{p}$, giving precisely one painted vertex ${\alpha }_2$. These statements concern complexified root spaces, not membership of $E_{ij}$ in real eigenspaces. Interchanging the labels of the two vertices gives the equivalent painting at ${\alpha }_1$.

In ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{R})$ put $H_1=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)$ and $H_2=\mathrm{diag}(1,1,-2)$. They commute, with $\theta H_1=H_1$ and $\theta H_2=-H_2$. For $Q=\mathrm{diag}(\left(\begin{array}{cc}1& 1\\ -i& i\end{array}\right),1)$, direct multiplication gives $Q^{-1}{H}_{1}Q=\mathrm{diag}(i,-i,0)$ and $Q^{-1}{H}_{2}Q=H_2$. Their complex span is therefore conjugate to the full diagonal Cartan. Its real part ${\mathfrak{h}}_0=\mathbb{R}{H}_1\oplus \mathbb{R}{H}_2$ is abelian and self-normalizing by complexification, hence a theta-stable real Cartan. In these diagonal coordinates put ${\beta }_1={\epsilon }_1-{\epsilon }_2$ and ${\beta }_2={\epsilon }_2-{\epsilon }_3$. On $H=x{H}_1+y{H}_2$, the positive-root values in the standard ordering are $2ix$, $-ix+3y$, and $ix+3y$. None vanishes identically on $\mathbb{R}{H}_1$, so no root is real and this Cartan is maximally compact by [L3].

The root action is $\theta {\beta }_1={\beta }_1$, $\theta {\beta }_2=-({\beta }_1+{\beta }_2)$ and $\theta ({\beta }_1+{\beta }_2)=-{\beta }_2$. Choose positive roots $\{{\beta }_1,-{\beta }_2,{\beta }_1+{\beta }_2\}$, all positive on $-i{H}_1\in i{\mathfrak{t}}_0$, so the system is compact-first compatible. Its simple roots are ${\delta }_1=-{\beta }_2$, ${\delta }_2={\beta }_1+{\beta }_2$, with ${\delta }_1+{\delta }_2={\beta }_1$. The involution exchanges ${\delta }_1$ and ${\delta }_2$. Thus this is an $A_2$ diagram with vertex interchange and no fixed vertex to paint. In particular the orbit containing ${\beta }_2$ is $\{{\beta }_2,-({\beta }_1+{\beta }_2)\}$, not the pair of positive roots in the incompatible standard ordering.

These three diagrams represent distinct equivalence classes, not merely different drawings. The empty painting with identity involution admits no painted reflection and stays empty under isomorphisms. A painted reflection retains its reflected vertex as painted, so cannot take a nonempty painting to the empty painting. Such a reflection commutes with the root involution, and relabeling conjugates its vertex permutation, so cannot turn the identity permutation into the interchange. Thus all three classes are different under the exact moves in [L4]. Isomorphic real forms would have equivalent diagrams by [L4], proving the pairwise non-isomorphism. The identity-involution empty-painting form is compact by step 1.1; in the intermediate form, $E_{13}+E_{31}$ is a real element with positive Killing square, so it is noncompact. All computations are in rank two, and AC is inherited only through the stated classification and Cartan interfaces.