Exercise 4, § 9

Suppose that K is ultrametric and $p > 0$ (in the notation of § 7). Let G be a finite-dimensional Lie group, A a group of automorphisms of G and B the corresponding group of automorphisms of $L(G)$. Let $G^A$ (resp. $L(G)^B$) be the set of elements of G (resp. $L(G)$) which are fixed under A (resp. B). Then $G^A$ is a Lie subgroup of G with Lie algebra $L(G)^B$. (Use the logarithmic mapping.)