Corollary 3
Tag 0AAL, permanent. Cite it as [Bourbaki, Tag 0AAL].
Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra
Let G, G', H be Lie groups and φ : G → H and φ' : G' → H Lie group morphisms. Suppose that K is of characteristic 0 and that H is finite-dimensional. Let F be the set of (g, g') ∈ G × G' such that φ(g) = φ'(g'). Then F is a Lie subgroup of G × G' and L(F) is the set of (x, x') ∈ L(G) × L(G') such that L(φ)x = L(φ')x'.
We apply Corollary 1 to the morphisms (g, g') ↦ φ(g) and (g, g') ↦ φ'(g') of G × G' into H.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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