Corollary 2

Tag 0AEO, permanent. Cite it as [Bourbaki, Tag 0AEO].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let G be a Lie group and H_1 and H_2 integral subgroups of G. Suppose that the topology on H_1 admits a countable base. Then

H_2 \subset H_1 \Leftrightarrow L(H_2) \subset L(H_1)

and, if these conditions hold, H_2 is an integral subgroup of H_1.

The last assertion and the implication $L(H_2) \subset L(H_1) \Rightarrow H_2 \subset H_1$ follow from Corollary 2 to Proposition 1. The converse implication follows from Proposition 3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 3, § 6
Proposition 11, § 9
Proposition 9, § 9

Cites

Corollary 2 to Proposition 1
Proposition 3