TITLE: connected compact semisimple lie group finite fundamental group
QUESTION [7 upvotes]: I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact.  Is there any source however that fleshes this out in detail / are there several ways to prove this fact?
Thanks!
(The result is often known as Weyl's theorem, I think for his take on the proof, Knapp provides a fairly detailed exposition of his perspective.)

REPLY [4 votes]: Every connected Liegroup, which has a semisimple Liealgebra with a definite Killing form is compact.
The Liealgebra of a compact Liegroup is always the direct sum of an semisimple and abelian Liealgebra, where the killing form of the semisimple part is negative definite.
So we can conclude, that the universal cover of your liegroup is compact and the finiteness of the fundamentalgroup follows immediately.
The fundamentalgroup is even always abelian.
The first two things can be found nicely in Serre's "Lie Algebras and Lie Groups" in Theorem 6.2 and 6.3.