Remark 2

Tag 01AL, permanent. Cite it as [Bourbaki, Tag 01AL].

Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups

With the notations of Prop. 4, every closed subgroup $H'$ of G containing H and such that $(H': H)$ is finite normalizes H, and hence is contained in N; similarly, the normalizer of $H'$ is contained in N. In particular, N is its own normalizer.

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

Corollary 1, § 4

Cites

Prop. 4