Corollary 2
Tag 019K, permanent. Cite it as [Bourbaki, Tag 019K].
Lie Groups and Lie Algebras, chapter IX, § 1. Compact Lie algebras
The centre and the fundamental group of a connected compact semi-simple Lie group are finite. Its universal covering is compact.
With the notation in Prop. 4, the groups V and T are reduced to the identity element; thus S is a universal covering of G, and the fundamental group of G is isomorphic to Ker , hence finite. The centre D of G is discrete since G is semi-simple, so D is finite.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.