Proposition 8

Tag 0A9J, permanent. Cite it as [Bourbaki, Tag 0A9J].

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

Let G, H be Lie groups and $\phi$ a morphism of G into H. If $t \in \mathscr{T}^{(\infty)}(G)$ , then $\phi_*(t^\vee) = (\phi_*(t))^\vee$ .

Let $\theta$ (resp. $\theta'$ ) be the mapping $g \mapsto g^{-1}$ of G into G (resp. of H into H). Then $\phi \circ \theta = \theta' \circ \phi$ , whence $\phi_*(\theta_*(t)) = \theta'_*(\phi_*(t))$ .

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

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