Corollary 1
Tag 0AEN, permanent. Cite it as [Bourbaki, Tag 0AEN].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let G be a finite-dimensional Lie group and H an integral subgroup of G. Then the Lie subalgebra tangent to H at e (§ 4, no. 5, Definitions 2 and 3) is L(H) and the Lie group structure on H is the structure induced by that on G.
As H is connected and finite-dimensional, its topology admits a countable base.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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