Proposition 1

Let K' be a non-discrete closed subfield of K, G a Lie group over K, V a manifold over K' and $(v, g) \mapsto vg$ a K'-*analytic mapping of* V $\times$ G into G. Suppose that, for all $v \in V$, the mapping $g \mapsto vg$ of G into G is an automorphism of G. Let $\varepsilon$ be an element of V such that $\varepsilon g = g$ for all $g \in G$ and $a \in T_\varepsilon(V)$. Then the vector field $g \mapsto ag$ on G is an infinitesimal automorphism of G.

For $v \in V,\ g_1 \in G,\ g_2 \in G,\ v(g_1g_2) = (vg_1)(vg_2)$. Hence, for $u_1 \in TG,\ u_2 \in TG,\ a(u_1u_2) = (au_1)(au_2)$ (§ 2, no. 1, Proposition 3). In particular, the mapping $g \mapsto ag$ of G into TG is a group homomorphism. On the other hand, this mapping is analytic by Lemma 2.