Proposition 15

Tag 0A9R, permanent. Cite it as [Bourbaki, Tag 0A9R].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let G (resp. G') be a Lie group and X (resp. X') a manifold of class C^r. Suppose that a law of left operation of class C^r of G (resp. G') on X (resp. X') is given. Let $\phi$ be a morphism of G into G' and $\psi$ a $\phi$ -morphism of X into X'. Let $t \in \mathscr{T}^{(s)}(G)$ , $u \in T^{(s')}(X)$ be such that $s + s' \leq r$ . Then

\psi_*(t * u) = \phi_*(t) * \psi_*(u).

This can be proved as is Proposition 6 of no. 2.

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

Cited by

§ 3. Passage from a Lie group to its Lie algebra
Proposition 21, § 3

Cites

Proposition 6