First and second homotopy groups of a connected Lie group

Question

I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$?

Thanks for anyone who give me references for a 'simple proof' of these results

Answer 1

Some references:

• This MO post describes the Eckmann-Hilton argument.
• Here's a blog post about abelian $\pi_1$.
• This other MO post has answers concerning the nullity of the second fundamental group.