Is there a simple proof (short and low-tech) of the following fact:

(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compact subgroup of $G$. Moreover, all maximal compact subgroups of $G$ are conjugate to $K$.

One nice corollary of this is that the group $K$ is a deformation retract of $G$ so these two groups have the same homotopy type. In the case where $G$ is semi-simple one may deduce the first part of the result above using the existence of the Iwasawa decomposition which is not so trivial to prove! But here since I'm only asking for a diffeomorphism between two manifolds, there might be a simpler argument!