Proposition 10
Tag 0AIW, permanent. Cite it as [Bourbaki, Tag 0AIW].
Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group
Let G be a finite-dimensional Lie group, g its Lie algebra and a a vector subspace of g. Then is a Lie subgroup of G with Lie algebra . This follows from § 3, Proposition 44 and Corollary 1 to Proposition 39.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.