Corollary 2
Tag 0AIK, permanent. Cite it as [Bourbaki, Tag 0AIK].
Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group
Let G be a Hausdorff topological group and A a subgroup of G. For A to be solvable (resp. nilpotent, commutative, it is necessary and sufficient that be so.
§ 9.2
This follows immediately from Proposition 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.