Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}\_n$. Does $G$ have finitely many connected components? I heard it's true when $\mathbf G$ is semisimple, by a theorem of Cartan. 



Real points of reductive groups and connected components
For every $\mathbb{R}$-scheme $X$ of finite type, $\pi\_0(X(\mathbb{R}))$ is finite. This follows e.g. from Theorem 2.3.6 in 
Bochnak, Coste, Roy, *Real Algebraic Geometry* 
(basic structure theorem for semi-algebraic sets).


