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I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group over $\mathbb R$. Assume that $\mathbf G$ is semisimple, i.e. $\mathbf G \times_{\mathbb R} \operatorname{Spec} \mathbb C$ is semisimple as an algebraic group (is Zariski-connected and has trivial radical). This implies that $\mathbf G$ is also connected in the Zariski topology.

Let $G = \mathbf G(\mathbb R)$, which is a real Lie group.

  • $G$ is in general not connected, right? For example, $\mathbf G = \operatorname{SO}(p,q)$. Does $G$ have finitely many components?

  • A real Lie group is defined to be semisimple if it is connected and if its Lie algebra is semisimple (has nondegenerate Killing form). The connected component $G^0$ of $G$ is a semisimple real Lie group, right?

  • Let $\mathfrak g$ be the Lie algebra of the algebraic group $\mathbf G$. Then do we have $\mathfrak g(\mathbb R) = \operatorname{Lie}(G)$?

  • Let $\theta$ be a Cartan involution on $\operatorname{Lie}(G)$. There is a corresponding involution Lie group automorphism $\Theta$ of $G^0$ whose differential is $\theta$, as in the Wikipedia article on Cartan decomposition. Does $\Theta$ extend to an automorphism of $G$?

  • Let $K$ be the maximal compact subgroup of $G^0$ obtained as the fixed points of $\Theta$. Is there a way to realize $K$ as the intersection with $G^0$ of a canonical maximal compact subgroup of $G$?

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In a slightly different order:

  • For future visitors to this problem, here an answer about connectivity on MO.

  • Yes, the functor from real linear algebraic groups to real Lie groups respects Lie algebras. See these notes by Milne, section III.2.

  • In characteristic zero, a linear algebraic group is semisimple iff its Lie algebra is semisimple. By the above point, $G^0$ is a semisimple real Lie group. (See the same notes, section II.4)

  • One can include a fixed choice of Cartan involution in the defintion of reductive real Lie group, and hence in the definition of semisimple real Lie group (as a reductive group with finite centre). A good choice is inverse-conjugate transpose, and then $K$ is essentially just the intersection of $G^0$ with the relevant orthogonal or unitary group, all as subgroups of some ambient $\mathrm{GL}(n,\mathbb{R})$. This is the approach taken by Knapp in his book Representation Theory of Semisimple Groups. This simplification might help answer the last part of your question.

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