Proposition 4
Tag 01AJ, permanent. Cite it as [Bourbaki, Tag 01AJ].
Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups
Let G be a compact Lie group, H a connected closed subgroup of G of maximal rank and N the normalizer of H in G. Then H is of finite index in N and is the identity component of N.
Indeed, the Lie algebra of H contains a Cartan subalgebra of L(G). Thus, by Chap. VII, §2, no. 1, Cor. 4 of Prop. 4, H is the identity component of N. Since N is compact, H is of finite index in N.
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.