Proposition 28
Tag 0AAH, permanent. Cite it as [Bourbaki, Tag 0AAH].
Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra
Let G and H be Lie groups and φ a morphism of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional.
(i) The kernel N of φ is a Lie subgroup of G and L(N) = Ker L(φ).
(ii) The morphism ψ of G/N into H derived from φ when passing to the quotient is an immersion.
(iii) If φ(G) is closed in H and the topology of G has a countable base, φ(G) is a Lie subgroup of H, ψ is an isomorphism of the Lie group G/N onto the Lie group φ(G) and L(φ(G)) = Im L(φ).
Let G operate on H on the left by the mapping (g, h) ↦ φ(g)h. It suffices to apply Proposition 14 of § 1, no. 7, to the orbit of e.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.