Vogan diagram for a fixed Cartan involution is well defined up to equivalence

Define $K=(\mathrm{Aut}({\mathfrak{g}}_0)\cap O(B_{\theta }){)}^0$. By [L2]--[L3] it is a compact connected Lie group. An automorphism preserves $B_{\theta }$ exactly when it commutes with $\theta$, since it preserves the nondegenerate form $B$. Differentiating shows that its Lie algebra consists of $\mathrm{ad}X$ commuting with $\theta$, equivalently $\mathrm{ad}(X-\theta X)=0$. Centerlessness gives $\mathrm{Lie}K=\mathrm{ad}{\mathfrak{k}}_0$. Conversely exponentials of these derivations preserve both forms. A connected group is generated by an exponential neighborhood, since the generated subgroup and its other cosets are open. Thus every element of $K$ is a product of $\exp (\mathrm{ad}X)$ with $X\in {\mathfrak{k}}_0$, is real inner, and commutes with $\theta$. This construction requires no finite-center integration of ${\mathfrak{g}}_0$.

Write ${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$. It has no real roots by [L1]. A root vanishes on ${\mathfrak{t}}_0$ precisely when it is real, so [L5] gives $Z_{{\mathfrak{g}}_0}({\mathfrak{t}}_0)={\mathfrak{h}}_0$ and $Z_{{\mathfrak{k}}_0}({\mathfrak{t}}_0)={\mathfrak{t}}_0$. For completeness, the complex Cartan assertion uses the nilpotence and self-normalization of the real Cartan: compact adjoints are skew-adjoint and split adjoints self-adjoint for $B_{\theta }$, so on the invariant Cartan their nilpotent restrictions are zero. Thus the real Cartan is abelian; comparison of real and imaginary parts shows its complexification is self-normalizing, hence a complex Cartan. The same adjoint identities on the ambient algebra make root values imaginary on ${\mathfrak{t}}_0$ and real on ${\mathfrak{a}}_0$. In particular ${\mathfrak{t}}_0$ is maximal abelian in ${\mathfrak{k}}_0$. The same statements hold for the second Cartan.

Identify ${\mathfrak{k}}_0$ with $\mathrm{Lie}K$ using injective $\mathrm{ad}$. For a maximal abelian subalgebra $\mathfrak{b}$, the closure of $\exp \mathfrak{b}$ in $K$ is compact, connected and abelian: exponentials commute by [L3], connectedness is preserved by closure, and continuity of commutators extends abelianness to the closure. Its Lie algebra contains $\mathfrak{b}$ and is abelian, so equals $\mathfrak{b}$. It is a maximal torus: a larger torus would give a larger abelian algebra, or would have the same algebra and contain the first as an open subgroup by exponential charts, forcing equality by connectedness. Apply maximal-torus conjugacy to the compact parts in step 1.2. It supplies $k\in K$ taking ${\mathfrak{t}}_0$ to ${\mathfrak{t}}_0^{\mathrm{\prime }}$, hence taking their centralizers ${\mathfrak{h}}_0$ to ${\mathfrak{h}}_0^{\mathrm{\prime }}$. The transported root map is $\alpha \mapsto \alpha \circ k^{-1}$, preserves Cartan integers, and intertwines theta and its eigenvalues on root lines. Thus it gives a diagram isomorphism from the first diagram to the diagram of the transported positive system on the second Cartan.

It remains to compare two compatible systems on one fixed Cartan ${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$. Put $E=i{\mathfrak{t}}_0$. Every root restricts nontrivially to $E$ by step 1.2. Each compatible system is realized by a point of $E$: choose a point $H$ realizing it in the real root space $i{\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$ of [L7] and average $H$ with $\theta H$. Every positive root is positive on both points, so the average remains in the same open chamber and lies in $E$. Consider the finitely many distinct hyperplanes $\ker (\alpha {\mathrm{\mid }}_E)$. Chambers of their complement therefore parametrize these systems. Any two such chambers can be joined by a finite sequence of adjacent chambers: choose interior endpoints and a generic polygonal path avoiding intersections of two distinct hyperplanes and crossing each hyperplane transversely. Such a path exists by avoiding finitely many proper affine subspaces when selecting intermediate vertices; it meets only finitely many walls. In dimension one there is just the wall $0$; in dimension zero there are no roots and nothing to prove.

Fix one crossing and a point $H_0$ of its wall lying on no other distinct wall. First suppose some nonzero weight $\gamma$ of ${\mathfrak{k}}_{\mathbb{C}}$ relative to ${\mathfrak{t}}_{\mathbb{C}}$ has this kernel. The torus from step 2.1 has Cartan algebra ${\mathfrak{t}}_{\mathbb{C}}$ in ${\mathfrak{k}}_{\mathbb{C}}$; thus $\gamma$ is a compact-group root. By [L4] an element of $K$ normalizing this torus acts on $E$ by a reflection fixing the wall pointwise. It normalizes ${\mathfrak{h}}_0=Z_{{\mathfrak{g}}_0}({\mathfrak{t}}_0)$, commutes with $\theta$, and permutes the full root restrictions. It therefore interchanges the two adjacent chambers: near $H_0$ it reverses the normal coordinate and fixes the wall, while no other hyperplane occurs locally. The resulting map of bases preserves all theta eigenvalues and Cartan integers. The diagrams are isomorphic, an allowed move of [L7].

If no such compact-group root exists, every full root whose restriction has this wall is imaginary and noncompact. Indeed a compact imaginary root already provides a nonzero weight in ${\mathfrak{k}}_{\mathbb{C}}$. A complex root $\alpha$ provides one as well: if $0\mathrm{\ne }X\in {\mathfrak{g}}_{\alpha }$, then $X+\theta X\mathrm{\ne }0$, lies in ${\mathfrak{k}}_{\mathbb{C}}$, and has weight $\alpha {\mathrm{\mid }}_E=(\theta \alpha ){\mathrm{\mid }}_E$. This rules out both cases. Imaginary roots vanish on ${\mathfrak{a}}_0$, so proportional restrictions to $E$ mean proportional full roots. Reducedness leaves only $\alpha ,-\alpha$ on this wall. Orient $\alpha$ to be positive before crossing. It is simple: a decomposition into two positive roots would force both to vanish at $H_0$, because all positive roots are nonnegative there; the only such positive root is $\alpha$, an impossibility. Crossing changes the sign of just this pair. Hence the second system is $s_{\alpha }$ of the first: for another positive root, reflection changes only its simple-$\alpha$ coefficient, and some other positive simple coefficient remains, so the one-sign property in [L6] keeps it positive. The root $\alpha$ is fixed by theta and painted.

We verify the painting change in step 3.2. Normalize the root triple $e,f,h_{\alpha }$ of [L6]. Since $\alpha$ is imaginary noncompact, $\theta e=-e$, $\theta f=-f$, and $\theta h_{\alpha }=h_{\alpha }$. For another fixed simple root $\beta$, put $m=-\beta (h_{\alpha })\ge 0$. Its string is $\beta ,\beta +\alpha ,\dots ,\beta +m\alpha$: $\beta -\alpha$ is not a root by the one-sign property, and [L6] determines the upper endpoint. This string is an irreducible root-${\mathfrak{s}\mathfrak{l}}_2$ module. To see this explicitly, all its weight spaces are one-dimensional by [L5]; the irreducible summand intervals in [L6] are centered at the same weight zero and overlap if more than one occurs, contradicting one-dimensionality. Thus $(\mathrm{ad}e{)}^m$ maps a nonzero $X_{\beta }$ to a nonzero vector of weight $s_{\alpha }\beta$. Applying theta to this iterated bracket gives eigenvalue $(-1{)}^m\epsilon (\beta )$. The color of $-\alpha$ stays painted, and nonfixed vertices remain paired because $s_{\alpha }$ commutes with theta. Relabel the new simple roots $s_{\alpha }\beta$ by their old positions; Cartan integers and the involution are unchanged. Colors of the other fixed vertices toggle exactly for odd $\beta (h_{\alpha })$, which is precisely $F_{\alpha }$ in [L7].

By step 2.2 a finite sequence of wall crossings connects the two compatible systems. Each crossing is an allowed diagram isomorphism by step 3.1 or an allowed painting move by steps 3.2 and 4.1. Combining this sequence with the Cartan conjugacy isomorphism of step 2.1 proves the claimed equivalence. The empty diagram of the zero algebra uses a sequence of zero moves. Compact factors cause no exception: their wall crossings are all of the first kind. Throughout theta is fixed, as required by the Statement.