Proposition 5

Tag 0A92, permanent. Cite it as [Bourbaki, Tag 0A92].

Lie Groups and Lie Algebras, chapter III, § 2. Group of tangent vectors to a Lie group

*Let $G$ and $H$ be Lie groups and $f$ a morphism of $G$ into $H$ . Then $T(f)$ is a morphism of the Lie group $T(G)$ into the Lie group $T(H)$ .

We know that $T(f')$ is analytic. On the other hand, let $m$ (resp. $n$ ) denote the multiplication on $G$ (resp. $H$ ). Then $f \circ m = n \circ (f \times f)$ , whence

T(f) \circ T(m) = T(n) \circ (T(f) \times T(f)),

which expresses the fact that $T(f)$ is a group homomorphism.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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