Corollary 2

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

Lie Groups and Lie Algebras, chapter III, § 10. The automorphism group of a Lie group

Let G be a semi-simple connected real or complex Lie group. The group Int G is the identity component of Aut G.

The mapping $u \mapsto L(u)$ is an isomorphism of Aut G onto a Lie subgroup of Aut $L(G)$ (Theorem 1). The image of Int G under this isomorphism is Ad G. But Ad G is the identity component of Aut $L(G)$ ( $§ 9$ , no. 8, Proposition 30 (ii)).

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

Cited by

Proposition 1, § 1
Corollary 5, § 1

Cites

Theorem 1