Maximal split abelian subspace and real rank

Assume the Axiom of Choice. Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with a Cartan involution $\theta$ and Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ (Cartan involution of a real semisimple Lie algebra, Cartan decomposition of a real semisimple Lie algebra). A maximal split abelian subspace of ${\mathfrak{g}}_0$ is a subspace $\mathfrak{a}\subseteq {\mathfrak{p}}_0$ that is abelian for the bracket, that is $[\mathfrak{a},\mathfrak{a}]=0$, and is maximal with this property among subspaces of ${\mathfrak{p}}_0$; equivalently (since ${\mathfrak{p}}_0$ is finite-dimensional, every abelian subspace of ${\mathfrak{p}}_0$ is contained in a maximal one) $\mathfrak{a}$ is a maximal abelian subspace of ${\mathfrak{p}}_0$. The real rank of ${\mathfrak{g}}_0$ is

${\mathrm{rank}}_{\mathbb{R}}{\mathfrak{g}}_0:=\dim \mathfrak{a},$

which is independent of the choice of $\mathfrak{a}$. The Lie-algebra form of the noncompact symmetric-pair construction in Riemannian symmetric pair of noncompact type applies to the pair $({\mathfrak{g}}_0,\theta )$ after its compact ideals are split off: those ideals lie in ${\mathfrak{k}}_0$ and contribute nothing to ${\mathfrak{p}}_0$, while the remaining ideal ${\mathfrak{g}}_{nc}$ has no compact ideal. This Lie-algebra datum does produce a symmetric pair. Namely, let $G=\mathrm{Aut}({\mathfrak{g}}_{nc}{)}^0$. Its Lie algebra is $\mathrm{Der}({\mathfrak{g}}_{nc})=\mathrm{ad}{\mathfrak{g}}_{nc}$, and centerlessness identifies this with ${\mathfrak{g}}_{nc}$ (Lie algebra of the automorphism group, Derivations of semisimple Lie algebras are inner, Semisimple Lie algebras are centerless and perfect). The center of $G$ is trivial: a central automorphism commutes with every $e^{t\mathrm{ad}X}$, so differentiation gives $\mathrm{ad}(AX)=\mathrm{ad}X$ for every $X$, and injectivity of $\mathrm{ad}$ gives $A=1$. Conjugation $A\mapsto \theta A\theta$ is a global involution of $G$ whose differential is $\theta$ under this identification. Thus $(G,G^{\mathrm{\Theta }})$ is the required noncompact-type symmetric pair, and Maximal abelian subspaces of p are conjugate by K ↗ conjugates any two maximal abelian subspaces of its ${\mathfrak{p}}_0$. They therefore have equal dimension. This downstream theorem is the well-definedness justification recorded in justified_by.

The terminology is related to the split real forms of Split real form: a real form ${\mathfrak{g}}_0$ is split precisely when ${\mathrm{rank}}_{\mathbb{R}}{\mathfrak{g}}_0$ equals the complex rank of the complexification ${\mathfrak{g}}_0^{\mathbb{C}}$, equivalently when a maximal abelian subspace $\mathfrak{a}\subseteq {\mathfrak{p}}_0$ (any one, by Maximal abelian subspaces of p are conjugate by K ↗) is a Cartan subalgebra of ${\mathfrak{g}}_0$; in general ${\mathrm{rank}}_{\mathbb{R}}{\mathfrak{g}}_0\in \{0,1,\dots ,\mathrm{rank}{\mathfrak{g}}_0^{\mathbb{C}}\}$, with value $0$ exactly for compact ${\mathfrak{g}}_0$.