Proposition 6

Let G be a Lie group germ, L its Lie algebra and $\phi : V \to G$ an exponential mapping of G. For all $x \in V$, let $T_x(L)$ be identified with L, so that the right differential $\varpi(x)$ of $\phi$ at x is a linear mapping of L into L. For x sufficiently close to 0,

Let L be given a norm compatible with its topology and such that $\| [x, y] \| \leq \| x \| \| y \|$ for all $x, y \in L$. It suffices to consider the case where G is the Lie group germ defined by L and $\phi = \operatorname{Id}_G$. By definition, $\varpi(x)$ is then the tangent mapping at x to the mapping $y \mapsto y . x^{-1}$ of G into G. If $H(X, Y)$ denotes the Hausdorff series, $\varpi(x)$ is therefore, for $\| x \|$ sufficiently small, the tangent mapping at 0 to the mapping $y \mapsto H(x, +y, -x)$ of G into G. In $H(X + Y, -X)$, the sum of terms of the first degree in Y is

(Chapter II, § 6, no. 5, Proposition 5). The proposition then follows from Differentiable and Analytic Manifolds, R, 3.2.4 and 4.2.3.

Let G be a Lie group germ and $t \in K$. A t-th power mapping of G is any mapping, defined and analytic on an open neighbourhood of e, with values in G and coinciding on a neighbourhood of e with a mapping

where $\phi$ is an injective exponential mapping of G.