Proposition 32

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

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

Let G and H be Lie groups, k a non-discrete closed subfield of K and $\phi$ a morphism of G into H for the Lie group structures over k. Suppose that K is of characteristic 0. If $L(\phi)$ is K-linear, $\phi$ is a morphism for the Lie group structures over K.

For all $g \in G$ ,

T_g(\phi) = T_e(\gamma(\phi(g))) \circ L(\phi) \circ T_g(\gamma(g)^{-1})

and hence $T_g(\phi)$ is K-linear. The proposition then follows from Differentiable and Analytic Manifolds, R, 5.14.6.

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

Cited by

Corollary 2, § 4

Cites

Nothing.