Theorem 1

Tag 019H, permanent. Cite it as [Bourbaki, Tag 019H].

Lie Groups and Lie Algebras, chapter IX, § 1. Compact Lie algebras

(H. Weyl) Let G be a connected Lie group whose Lie algebra is compact semi-simple. Then G is compact and its centre is finite.

Since G is semi-simple, its centre D is discrete. Moreover, the quotient group $G/D$ is isomorphic to Ad(G) (Chap. III, §6, no. 4, Cor. 4), hence compact (Prop. 3). Finally, the group $G/D$ is equal to its derived group (Chap. III, §9, no. 2, Cor. of Prop. 4). The theorem now follows from Integration, Chap. VII, §3, no. 2, Prop. 5.

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.

Cited by

Proposition 4, § 1
Proposition 16, § 4

Cites

Integration, Chap. VII, §3, no. 2, Prop. 5 not in this corpus: INT VII
Chap. III, §6, no. 4, Cor. 4 not in this corpus
Chap. III, §9, no. 2, Cor. of Prop. 4 not in this corpus
Prop. 3