I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps:

- Take one maximal compact subgroup $K$ of $G$ (which I know that exists). Consider $G/K$. This is a $CAT(0)$ space.
- Take any other maximal compact subgroup $L$ of $G$. $L$ acts on $G/K$ so by Cartan's fixed point theorem (or something similar), $L$ has a fixed point in $G/K$, call it $x_0$.
- The stabilizer of $x_0$ in $G$ is a conjugate of $K$, so $L\subseteq gKg^{-1}$ for some $g\in G$. By maximality of $L$ we get an equality, concluding the proof.

To prove the first stage one defines a Riemannian structure on $G/K$, which turns out to have non-negative sectional curvature $K(X,Y)=-||[X,Y]||^2$, so it is locally $CAT(0)$. However, I could not see why $G/K$ is simply connected.

I could not find a reference giving a neat (or any) proof of that fact. I would be very grateful if someone could provide me with a reference for this fact.

Thanks in advance, Miel.