Proposition 8

Let G and H be Lie group germs, h a morphism of G into H and $\phi_G$ and $\phi_H$ exponential mappings of G and H. There exists a neighbourhood V of 0 in $L(G)$ such that $h \circ \phi_G$ and $\phi_H \circ L(h)$ coincide on V.

Let $L(G)$ and $L(H)$ be given norms defining their topologies and such that $\| [x, y] \| \leq \| x \| \| y \|$ for all x and y. It can be assumed that G (resp. H) is the Lie group germ defined by $L(G)$ (resp. $L(H)$), so that $\phi_G$ (resp. $\phi_H$) coincides with $\mathrm{Id}_G$ (resp. $\mathrm{Id}_H$) in a neighbourhood of 0. On the other hand, there exists a symmetric open neighbourhood W of 0 in $L(G)$ such that $L(h)$ is a morphism of the Lie group germ W into H. By Theorem 1, $L(h)$ coincides with h on a neighbourhood of 0, whence the proposition.

Loosely speaking, if G and H are identified in a neighbourhood of the identity element with $L(G)$ and $L(H)$ by means of exponential mappings, every morphism of G into H is linear in a neighbourhood of 0.