Cohomology of Lie Groups finite dimensional?

Question

Let $G$ be a connected Lie group, without any further assumptions. Is it true, that its rational cohomology ring $$H^\bullet(G,\mathbb Q)$$ is finite dimensional? Is $G$ homotopy equivalent to a compact Lie group?

Answer 1

Both questions have the answer of "yes". Of course, an answer of "yes" to the second must imply and answer of "yes" to the first because every compact manifold has a finitely generated cohomology ring.

So, why is $G$ homotopy equivalent to a compact Lie group? In fact, more is true. We have the following theorem (see wiki for more):

Suppose $G$ is a connected subgroup. Then, there exists a maximal compact subgroup $K\subseteq G$. Further, all such maximal compact subgroups are conjugate and $G$ is diffeomorphic to $K\times \mathbb{R}^n$ for some $n$.

(Note that while $G$ is diffeomorphic to $K\times\mathbb{R}^n$, $G$ is only rarely isomorphic (as a group) to a product of $K$ and $\mathbb{R}^n$.)

Finally, simply note that $K\times\mathbb{R}^n$ obviously deformation retracts onto $K$.