Exercise 3, § 4

Let G be a Lie group, U an open neighbourhood of 0 in L(G) and $\phi$ an analytic mapping of U into G such that $\phi(0) = e$ and $T_0 \phi = \mathrm{Id}_{L(G)}$. The following conditions are equivalent: (a) $\phi$ is an exponential mapping; (b) There exists $m \in \mathbf{Z}$ distinct from 0, 1, -1 such that $\phi(mx) = \phi(x)^m$ in a neighbourhood of 0. (If condition (b) holds, show that $\phi^*$ satisfies condition (iii) of Theorem 4.)