Corollary 2
Tag 0AAK, permanent. Cite it as [Bourbaki, Tag 0AAK].
Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra
Let G be a finite-dimensional Lie group and G₁ and G₂ two Lie subgroups of G. Suppose that K is of characteristic 0. Then G₁ ∩ G₂ is a Lie subgroup of G with Lie algebra L(G₁) ∩ L(G₂).
We apply Proposition 29 to the canonical injection of G₁ into G and the subgroup G₂.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.