Proposition 8

Tag 0AF2, permanent. Cite it as [Bourbaki, Tag 0AF2].

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

*Let $G$ be a Lie group. For all $x, y$ in $L(G)$ and $n$ an integer,

\exp(x + y) = \lim_{n \to + \infty} \left( \left( \exp \frac{1}{n} x \right) \left( \exp \frac{1}{n} y \right) \right)^n

\exp[x, y] = \lim_{n \to + \infty} \left( \left( \exp \frac{1}{n} x \right) \left( \exp \frac{1}{n} y \right) \left( \exp \frac{1}{n} x \right)^{-1} \left( \exp \frac{1}{n} y \right)^{-1} \right)^{n^2}.

By Proposition 7, this follows from Proposition 4 of § 4, no. 3, taking $\lambda = \frac{1}{n}$ .

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

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