Cayley transforms connect theta-stable Cartans in the classification

Invariance gives $(\mathrm{ad}X{)}^{\ast }=-\mathrm{ad}(\theta X)$ for $X\in {\mathfrak{g}}_0$ and the inner product $-B(\cdot ,\theta \cdot )$. On a theta-stable Cartan, each $\mathrm{ad}T$ for $T\in {\mathfrak{t}}_0$ is skew-adjoint and each $\mathrm{ad}A$ for $A\in {\mathfrak{a}}_0$ is self-adjoint. Their restrictions to that invariant subspace are nilpotent by nilpotence of the Cartan. They are diagonalizable by [L5], so those restrictions vanish. Thus the entire Cartan is abelian. Its complexification is nilpotent and self-normalizing: if $X+iY$ normalizes it, comparison of real and imaginary parts against ${\mathfrak{h}}_0$ shows that $X,Y$ normalize ${\mathfrak{h}}_0$ and belong to it. Hence it is a complex Cartan and [L2] applies. The same adjoint identities on the ambient algebra imply that roots are imaginary on ${\mathfrak{t}}_0$ and real on ${\mathfrak{a}}_0$.

We construct the compact group needed for conjugacy without any assumption on a chosen global cover. Let $K$ be the identity component of $\mathrm{Aut}({\mathfrak{g}}_0)\cap O(B_{\theta })$. This intersection is a closed subgroup of the compact orthogonal group by [L8], so $K$ is a compact connected Lie group by [L6]. An automorphism preserves $B$ by invariance of trace under change of basis; therefore it preserves $B_{\theta }$ exactly when it commutes with $\theta$. Its tangent algebra consists of $\mathrm{ad}X$ commuting with $\theta$, equivalently $\mathrm{ad}(X-\theta X)=0$, so it is $\mathrm{ad}{\mathfrak{k}}_0$ by centerlessness. Conversely the exponentials of these derivations lie in the intersection. A connected Lie group is generated by an exponential identity neighborhood: the subgroup so generated is open, and its other cosets are open, so connectedness makes it the whole group. Thus all members of $K$ are products of $\exp (\mathrm{ad}X)$, $X\in {\mathfrak{k}}_0$, and are real inner automorphisms. They preserve both ${\mathfrak{k}}_0$ and ${\mathfrak{p}}_0$.

Put ${\mathfrak{h}}_{\mathbb{R}}=i{\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$. The form $B$ is real positive definite there, and every root is real on it by step 1.1. Thus its Killing-dual vector belongs to this real subspace. For a real root $H_{\alpha }\in {\mathfrak{a}}_0$, while for an imaginary root $H_{\beta }\in i{\mathfrak{t}}_0$, by orthogonality and the respective vanishing conditions. In both cases $\theta \sigma H_{\gamma }=-H_{\gamma }$, so $\mathrm{\mid }\gamma {\mathrm{\mid }}^2=B(H_{\gamma },H_{\gamma })>0$ and $h_{\gamma }=2H_{\gamma }\mathrm{/}\mathrm{\mid }\gamma {\mathrm{\mid }}^2$. In particular $B(iT,iT)=-B(T,T)>0$ for nonzero $T\in {\mathfrak{t}}_0$; imaginary roots do not introduce a negative coroot sign. Bracketing root vectors shows $\theta {\mathfrak{g}}_{\gamma }={\mathfrak{g}}_{\theta \gamma }$ and $\sigma {\mathfrak{g}}_{\gamma }={\mathfrak{g}}_{-\theta \gamma }$. An imaginary root space is a one-dimensional theta eigenspace.

In any compact connected Lie group $C$, a maximal abelian Lie subalgebra $\mathfrak{b}$ is the Lie algebra of a maximal torus. Indeed the closure of $\exp \mathfrak{b}$ is connected, compact and abelian, with Lie algebra containing $\mathfrak{b}$; abelianness and maximality give equality. Any larger torus has larger abelian Lie algebra, unless it has the same algebra, in which case the exponential neighborhood makes the original torus open and hence equal to the connected larger one. Conversely a maximal torus has maximal abelian Lie algebra by the same closure argument applied to an abelian enlargement. Here closure preserves connectedness, and continuity of the commutator extends commutativity to the closure. It follows from [L6] that maximal abelian Lie subalgebras of $C$ are conjugate. Apply this to $K$ of step 1.2, identifying its Lie algebra with ${\mathfrak{k}}_0$ by the injective map $\mathrm{ad}$; conjugation on $\mathrm{ad}{\mathfrak{k}}_0$ corresponds to the natural $K$-action on ${\mathfrak{k}}_0$.

Any two maximal abelian subspaces $\mathfrak{a},{\mathfrak{a}}^{\mathrm{\prime }}$ of ${\mathfrak{p}}_0$ are conjugate under $K$. The commuting self-adjoint maps $\mathrm{ad}A$, $A\in \mathfrak{a}$, have finitely many simultaneous real weights by [L5]. Choose $H\in \mathfrak{a}$ off the kernels of their nonzero weights; then $Z_{{\mathfrak{p}}_0}(H)=Z_{{\mathfrak{p}}_0}(\mathfrak{a})=\mathfrak{a}$, the last equality by maximal abelianness. Similarly choose $H^{\mathrm{\prime }}\in {\mathfrak{a}}^{\mathrm{\prime }}$. The function $k\mapsto B(k{H}^{\mathrm{\prime }},H)$ has an extremum on compact $K$. Differentiating there along $\exp (t\mathrm{ad}Z)$, for $Z\in {\mathfrak{k}}_0$, gives $B(Z,[k{H}^{\mathrm{\prime }},H])=0$. Since the bracket lies in ${\mathfrak{k}}_0$ and $B$ is nondegenerate there, it vanishes. Hence $k{H}^{\mathrm{\prime }}\in \mathfrak{a}$ and $\mathfrak{a}\subseteq Z_{{\mathfrak{p}}_0}(k{H}^{\mathrm{\prime }})=k{\mathfrak{a}}^{\mathrm{\prime }}$. Maximality makes this equality. Empty families of nonzero weights cause no difficulty: choose $H=0$, in which case the same centralizer assertion holds.

For opposite root vectors, invariance and [L3] give $[E,F]=B(E,F)H_{\gamma }$: pair with an arbitrary $H\in \mathfrak{h}$ to get $B([E,F],H)=\gamma (H)B(E,F)$ and use nondegeneracy in [L2]. If $\beta$ is noncompact imaginary, then $\sigma E\in {\mathfrak{g}}_{-\beta }$ and $B(E,\sigma E)=⟨E,E⟩>0$. If $\alpha$ is real, its sigma-stable root line has a nonzero real vector $E$ (take $v+\sigma v$ or $i(v-\sigma v)$), and $-B(E,\theta E)=⟨E,E⟩>0$. Positive real rescaling gives [L7]. Consequently $(E_{\beta },\sigma E_{\beta },h_{\beta })$ and $(E_{\alpha },-\theta E_{\alpha },h_{\alpha })$ satisfy $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$.

The root decomposition gives the centralizer of a subspace of $\mathfrak{h}$ by retaining precisely its vanishing roots. Thus if there is no real root, $Z_{{\mathfrak{g}}_0}({\mathfrak{t}}_0)={\mathfrak{h}}_0$ and $Z_{{\mathfrak{k}}_0}({\mathfrak{t}}_0)={\mathfrak{t}}_0$. This makes ${\mathfrak{t}}_0$ maximal abelian in ${\mathfrak{k}}_0$, and by step 2.2 its dimension bounds the compact part of every theta-stable Cartan. If there is no noncompact imaginary root, the root spaces in $Z_{\mathfrak{g}}({\mathfrak{a}}_0)$ other than $\mathfrak{h}$ are compact imaginary and lie in $\mathfrak{k}$. Hence $Z_{{\mathfrak{p}}_0}({\mathfrak{a}}_0)={\mathfrak{a}}_0$, so ${\mathfrak{a}}_0$ is maximal abelian in ${\mathfrak{p}}_0$ and step 2.3 makes its dimension maximal among split parts. Abelian subspaces can always be enlarged to maximal ones by maximizing their bounded integer dimension.

For an imaginary root write $e=E_{\beta }$, $f=\sigma e$, $h=h_{\beta }$. The operator $D=(\pi \mathrm{/}4)\mathrm{ad}(f-e)$ sends $h$ to $(\pi \mathrm{/}2)(e+f)$ and $e+f$ to $-(\pi \mathrm{/}2)h$, and kills $e-f$. Therefore its exponential satisfies $c(h)=e+f$, $c(e+f)=-h$, $c(e-f)=e-f$. For a real root write $e=E_{\alpha }$, $f=-\theta e$, $h=h_{\alpha }$. The operator $D=-(i\pi \mathrm{/}4)\mathrm{ad}(e+f)$ sends $h$ to $(i\pi \mathrm{/}2)(e-f)$ and $e-f$ to $(i\pi \mathrm{/}2)h$, and kills $e+f$. Thus $d(h)=i(e-f)$, $d(e-f)=ih$, $d(e+f)=e+f$. In each two-dimensional plane the square of $D$ is $-(\pi \mathrm{/}2{)}^2$, so these formulas follow directly from the exponential series. Both transforms fix the relevant root kernel in $\mathfrak{h}$, since both root vectors commute with it.

Decompose $\mathfrak{h}=\ker \gamma \oplus \mathbb{C}{h}_{\gamma }$. In the imaginary case, $\ker \beta$ is the complexification of its kernel on ${\mathfrak{h}}_0$, and $e+\sigma e$ is nonzero, real and in ${\mathfrak{p}}_0$. Step 4.1 therefore gives ${\mathfrak{g}}_0\cap c(\mathfrak{h})=\ker (\beta {\mathrm{\mid }}_{{\mathfrak{h}}_0})\oplus \mathbb{R}(e+\sigma e)$. Its compact dimension decreases by one and its split dimension increases by one. In the real case the line $\mathbb{C}i(e+\theta e)=\mathbb{C}(e+\theta e)$ has real part $\mathbb{R}(e+\theta e)\subseteq {\mathfrak{k}}_0$, giving the stated kernel formula for $d$ and the opposite dimension changes. The new lines are independent of the kernels because the root decomposition separates them from $\mathfrak{h}$. These subspaces are theta-stable and their complexifications are exactly the transformed complex Cartans; they are abelian, and any real normalizer complexifies into that Cartan, so they are real Cartan subalgebras.

For a real root retain $e,f,h$ from step 4.1, so $f=-\theta e$ and all three are real. Put $\beta =d(\alpha )$ and $E^{\mathrm{\prime }}=-id(e)=(h-i(e+f))\mathrm{/}2$. This is a vector in the transported root line. Both $h$ and $e+f$ are in ${\mathfrak{p}}_0$, so $\theta E^{\mathrm{\prime }}=-E^{\mathrm{\prime }}$. The root $\beta$ vanishes on the new split part $\ker (\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0})$ and on the old compact kernel; on the new compact generator $e-f$ it has value $-2i$, because $d^{-1}(e-f)=-ih$. Thus it is noncompact imaginary. Its Killing-dual vector is $d{H}_{\alpha }$, whence $\mathrm{\mid }\beta {\mathrm{\mid }}^2=B(d{H}_{\alpha },d{H}_{\alpha })=\mathrm{\mid }\alpha {\mathrm{\mid }}^2$ by step 2.1 and preservation of $B$. Root orthogonality gives $B(h,e+f)=0$, $B(e+f,e+f)=2B(e,f)=4\mathrm{/}\mathrm{\mid }\alpha {\mathrm{\mid }}^2=B(h,h)$. Consequently $E^{\mathrm{\prime }}+\sigma E^{\mathrm{\prime }}=h$, $B(E^{\mathrm{\prime }},\sigma E^{\mathrm{\prime }})=2\mathrm{/}\mathrm{\mid }\alpha {\mathrm{\mid }}^2$ and $\sigma E^{\mathrm{\prime }}-E^{\mathrm{\prime }}=i(e+f)$. Hence $c_{\beta }=\exp (i\pi \mathrm{ad}(e+f)\mathrm{/}4)=d^{-1}$. The inverse operation on real Cartans is ${\mathfrak{g}}_0\cap c_{\beta }(({\mathfrak{g}}_0\cap d\mathfrak{h}{)}_{\mathbb{C}})={\mathfrak{g}}_0\cap \mathfrak{h}={\mathfrak{h}}_0$, using the complexification equality in step 5.1. No assertion that the complex automorphism $d$ preserves ${\mathfrak{g}}_0$ is needed.

Step 5.1 shows that a real root prevents maximal compactness and a noncompact imaginary root prevents maximal split dimension. Together with step 3.2 this proves both criteria: maximal compactness is equivalent to absence of real roots, and maximal split dimension to absence of noncompact imaginary roots. Repeated real-root transforms increase the integer compact dimension by one, bounded by $\dim {\mathfrak{k}}_0$; therefore they stop at a maximally compact Cartan. Independently, repeated noncompact-imaginary transforms increase split dimension by one, bounded by $\dim {\mathfrak{p}}_0$, and stop at a maximally noncompact Cartan.

For two maximally compact Cartans, step 6.2 and step 3.2 make their compact parts maximal abelian in ${\mathfrak{k}}_0$ and make each Cartan the centralizer of its compact part. Step 2.2 conjugates the compact parts by $K$, and thus conjugates those centralizers. For two maximally noncompact Cartans, step 6.2 and step 3.2 make their split parts maximal abelian in ${\mathfrak{p}}_0$; step 2.3 first conjugates these to a common $\mathfrak{a}$. Write the Cartans as $\mathfrak{t}\oplus \mathfrak{a}$ and ${\mathfrak{t}}^{\mathrm{\prime }}\oplus \mathfrak{a}$. Put $\mathfrak{m}=Z_{{\mathfrak{k}}_0}(\mathfrak{a})$. Each compact part is maximal abelian in $\mathfrak{m}$: a vector there commuting with $\mathfrak{t}$ also centralizes the whole Cartan and hence belongs to it by self-normalization, and its compact component belongs to $\mathfrak{t}$. The pointwise stabilizer $M=\{k\in K:kA=A\text{ for all }A\in \mathfrak{a}\}$ is closed and compact. Differentiating and exponentiating its defining equations shows $\mathrm{Lie}M=\mathrm{ad}\mathfrak{m}$. Apply step 2.2 to the compact connected group $M^0$: it conjugates $\mathfrak{t}$ to ${\mathfrak{t}}^{\mathrm{\prime }}$ while fixing $\mathfrak{a}$ pointwise. Thus it conjugates the full Cartans. All conjugations lie in $K$ and are real inner by step 1.2.

The kernel and dimension formulas of step 5.1 prove assertions 1 and 2; step 6.1 proves the compatible inverse assertion 3; steps 6.2 and 7.1 prove assertion 4. In the zero algebra there are no roots, both maximal dimensions are zero, and the empty sequence suffices. Compact factors and zero split part cause no exception to the compact-group constructions or bounded-dimension arguments.