Proposition 5
Tag 0A7I, permanent. Cite it as [Bourbaki, Tag 0A7I].
Lie Groups and Lie Algebras, chapter III, § 1. Lie groups
Let G be a Lie group and H a Lie subgroup of G. (i) H is closed in G. (ii) The canonical injection of H into G is a Lie group morphism. (iii) Let L be a Lie group and f a mapping of L into G such that f(L) ⊂ H. For f to be a morphism of L into H, it is necessary and sufficient that f be a morphism of L into G.
By Differentiable and Analytic Manifolds, R, 5.8.3, H is locally closed. Hence H is closed (General Topology, Chapter III, § 2, Proposition 4). Assertion (ii) is obvious. Assertion (iii) follows from Differentiable and Analytic Manifolds, R, 5.8.5.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.