Corollary 1
Tag 0AAJ, permanent. Cite it as [Bourbaki, Tag 0AAJ].
Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra
Let G, H be Lie groups and φ₁ and φ₂ morphisms of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional. The set of g ∈ G such that φ₁(g) = φ₂(g) is a Lie subgroup G' of G and L(G') is the set of x ∈ L(G) such that L(φ₁)x = L(φ₂)x.
We write φ(g) = (φ₁(g), φ₂(g)) for all g ∈ G, so that φ is a morphism of G into H × H. Let Δ be the diagonal subgroup of H × H. Then G' = φ⁻¹(Δ) and L(φ)x = (L(φ₁)x, L(φ₂)x) for all x ∈ L(G). It now suffices to apply Proposition 29.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.