TITLE: Homotopy groups of Lie groups
QUESTION [139 upvotes]: Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact.  Is there a nice argument, perhaps like a more clever version of the proof that $\pi_1(G)$ must be abelian?

REPLY [2 votes]: There is a sketch of the proof using the maximal torus as an exercise in Greub-Halperin-Vanstone's book.