Proposition 7

Tag 0A95, permanent. Cite it as [Bourbaki, Tag 0A95].

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

Let $G$ be a Lie group, $M$ a manifold of class $C^r$ and $f$ and $g$ mappings of class $C^r$ of $M$ into $G$ , so that $fg$ is a mapping of class $C^r$ of $M$ into $G$ . Let $m \in M, x = f(m), y = g(m), u \in T_m(M)$ . Then

(T fg)u = T(f)u.y + x.T(g)u.

Let $m$ be the multiplication of $G$ . Then $fg = m \circ (f, g)$ . Now

T(f, g)(u) = (T(f)u, T(g)u),

hence $T(fg)u = T(f)u.T(g)u$ . It then suffices to apply (1) with $f$ replaced by $m$ .

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

Cited by

Corollary, § 2
Proposition 59, § 3

Cites

Nothing.