Proposition 6

Tag 0AEY, permanent. Cite it as [Bourbaki, Tag 0AEY].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let G be a connected complex Lie group. If G is compact, G is commutative.

The holomorphic mapping $g \mapsto \mathrm{Ad}\,g$ of G into $\mathscr{L}(L(G))$ is constant (Differentiable and Analytic Manifolds, R, 3.3.7) and hence $\mathrm{ad}\,a = 0$ for all $a \in L(G)$ ( $§ 3$ , no. 12, Proposition 44). Hence G is commutative ( $§ 4$ , Corollary 3 to Theorem 1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Exercise 26, § 6
Exercise 1, § 1

Cites

Nothing.