Corollary 3

*Let $G$ be a Lie group, $\rho$ an analytic linear representation of $G$, $x \in L(G)$ and $g \in G$. (i) $\rho(\exp x) = \exp L(\rho)x$; (ii) $\mathrm{Ad}(\exp x) = \exp \mathrm{ad}\, x$; (iii) $g(\exp x)g^{-1} = \exp(\mathrm{Ad}\, g.x)$. The argument is as in $§ 4$, no. 4, Corollaries 2 and 3 to Proposition 8.