Last year I wrote up how to prove that the fundamental group of a (connected) topological group was abelian. Since Lie groups are topological groups, they also have abelian fundamental groups, but I think there is a much neater way to prove this fact using smooth things. Here it is:
Lemma 1: A connected Lie group that acts smoothly on a discrete space is must be a trivial action.
Proof: Suppose our action is by . Consider a point with non-trivial orbit. Then where is non-empty. Thus , a disconnection of . Thus all orbits are trivial.
Lemma 2: A discrete normal subgroup of a Lie group (from now on always assumed connected) is central.
Proof: Denote the subgroup . Then is a smooth action on (a discrete set). Thus by Lemma 1, for all . Thus is central, and in particular abelian.
Now for the main theorem. Let be a connected Lie group. Let be the universal cover of . Then the covering map is a group homomorphism. Since is simply connected, the covering is normal and hence . By virtue of being normal, we also get that acts transitively on the fibers of . In particular, on the set , which is discrete being the fiber of a discrete bundle. But this set is , which is a normal subgroup. I.e. a discrete normal subgroup, which by Lemma 2 is abelian.
Fix . Then we get an ismorphism by where is the unique covering automorphism that takes to . Thus is abelian which means is abelian.