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Let $G$ be a linear connected semisimple Lie group, $\mathfrak g$ its Lie algebra. With respect to the Cartan involution $$ \theta:X\mapsto -\overline{X}^t, $$ one has $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$.

Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak p$, $\mathfrak{t}$ a maximal abelian subspace of $\mathfrak{m}=C_\mathfrak{k}(\mathfrak a)$. Then $$\mathfrak{h}_r=\mathfrak{a}\oplus \mathfrak{t}$$ is a $\theta$-stable Cartan subalgebra with maxinum real rank.

Let $\mathfrak{t_0}$ be a maximal abelian subspace of $\mathfrak k$. Then $$\mathfrak h_1=C_\mathfrak{g}(\mathfrak{t}_0)=\mathfrak t_0\oplus C_\mathfrak{p}(\mathfrak{t}_0)$$ is the $\theta$-stable Cartan subalgebra with minumun real rank.

How to classify other $\theta$-stabl Cartan subalgebras?

Qinghua Pi
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  • I'm not sure if this addresses the question, but using Cayley transforms associated with real or imaginary non-compact roots, one can obtain all $\theta$-stable Cartan subalgebras of $\mathfrak{g}$, starting from any given one. A possible reference could be section 7 of chapter VI in the book A. W. Knapp, 'Lie groups beyond an introduction', 2nd ed., Birkhäuser, 2002 – ivanpenev Jun 02 '14 at 16:44

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