Proposition 12

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

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

*Let G be a Lie group and $L = L(G)$ . For all $x \in L$ , let $T_x(L)$ be identified with $L$ , so that the right differential $\varpi(x)$ of $\exp$ at $x$ is a linear mapping of $L$ into $L$ . For all $x \in L$ ,

\varpi(x) = \sum_{n \geqslant 0} \frac{1}{(n + 1)!} (\mathrm{ad}\ x)^n.

The two sides are analytic functions of $x$ and are equal for $x$ sufficiently close to 0 (§ 4, no. 3, Proposition 6).

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

Cited by

Corollary, § 6
Proposition 2, § 6

Cites

§ 4, no. 3, Proposition 6