Proposition 60

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

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

Let $G$ be a Lie group, of finite dimension if $K$ is of characteristic > 0, $M$ a manifold of class $C^r$ , $f$ a mapping of class $C^r$ of $M$ into $G$ and $\alpha$ the left differential of $f$ . Then $d\alpha + [\alpha]^2 = 0$ .

Let $\omega$ be the canonical left differential form of $G$ . Using Corollary 1 to Proposition 51, no. 14, we have

d\alpha = d(f^*(\omega)) = f^*(d\omega) = f^*(-[\omega]^2) = -[f^*(\omega)]^2 = -[\alpha]^2.

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

Cited by

Nothing.

Cites

Corollary 1 to Proposition 51