Proposition 4

Tag 0AH8, permanent. Cite it as [Bourbaki, Tag 0AH8].

Let G be a Lie group and $\phi$ an injective exponential mapping of G. Suppose that $p > 0$ . For all $x, y$ in $L(G)$ ,

(1)

x + y = \lim_{n \to +\infty} p^{-n} \phi^{-1}(\phi(p^n x) \phi(p^n y))

(2)

[x, y] = \lim_{n \to +\infty} p^{-2n} \phi^{-1}(\phi(p^n x) \phi(p^n y) \phi(-p^n x) \phi(-p^n y)).

These are special cases of Proposition 4 of § 4, no. 3.

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

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Nothing.