Proposition 19
Tag 0AJ8, permanent. Cite it as [Bourbaki, Tag 0AJ8].
Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group
Let be a finite-dimensional Lie group. For to be solvable, it is necessary and sufficient that possess a solvable open subgroup.
The proof is analogous to that of Proposition 12 of no. 5.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.