Corollary
Tag 0AEU, permanent. Cite it as [Bourbaki, Tag 0AEU].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let G be a finite-dimensional Lie group and H_1 and H_2 two integral subgroups of G. The subgroup of G generated by H_1 and H_2 and the subgroup (H_1, H_2) of G are integral subgroups of G.
The subgroup (G, G) of G is not always closed (§ 9, Exercise 6).
Recall (§ 3, no. 11, Corollary 5 to Proposition 41) that if a is a finite-dimensional algebra, Aut(a) is a Lie subalgebra of GL(a) and that L(Aut(a)) is the Lie algebra of derivations of a.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.