Proposition 3

Tag 0A6O, permanent. Cite it as [Bourbaki, Tag 0A6O].

Lie Groups and Lie Algebras, chapter II, § 7. Convergence of the Hausdorff series (real or complex case)

For $\|x\| + \|y\| < \frac{1}{2} \log 2$ , (23) $\exp_A x \cdot \exp_A y = \exp_A h(x, y).$ It follows from (18) and the relation $e^u e^v = e^{u+v}$ that

m \circ (1 + \tilde{e}, 1 + \tilde{e}) = (1 + \tilde{e}) \circ \tilde{H}

\hat{P}(A \times A; A)

. We therefore deduce from Differentiable and Analytic Manifolds, R, 3.1.9 that (23) is true for

(x, y)

sufficiently close to

(0, 0)

, whence the proposition follows by analytic continuation (Differentiable and Analytic Manifolds, R, 3.2.5).

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

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Cites

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