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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. $mathbf G times_mathbb R operatornameSpec mathbb C$ 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, $mathbf G = operatornameSO(p,q)$. 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 $mathfrak g(mathbb R) = operatornameLie(G)$?




• Let $theta$ be a Cartan involution on $operatornameLie(G)$. 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 $G$?

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 $mathrmGL(n,mathbbR)$. 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.