Corollary

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

Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group

If $a = n_g(a)$ , A is a Lie subgroup of G and is the identity component of $N_G(A)$ . This identity component is a Lie subgroup with Lie algebra $n_g(a)$ (Proposition 11) and hence is equal to A by § 6, no. 2, Theorem 2 (i).

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

Cited by

Corollary 4, § 2

Cites

§ 6, no. 2, Theorem 2
Proposition 11