Let $G$ be a complex semisimple Lie group and $H$ be a maximal compact subgroup.
Prove that $H^{\mathbb C} = G$
This is what I've thought about : I want to do it at the level of Lie algebra. Let $h = Lie(H)$ and $g = Lie(G)$. Let the Cartan decomposition of $g$ be $g = h+ m$. I somehow want to show that $h + ih$ is a Cartan decomposition. Then $h \otimes \mathbb C = g$ and we are done.
I'd appreciate any help available.