Corollary 2

Let k be a non-discrete closed subfield of K, G a Lie group over k and L = L(G). Suppose that L has a normable Lie K-algebra structure L', compatible with the normable Lie k-algebra structure and invariant under the adjoint representation of G. Then there exists on G one and only one Lie K-group structure compatible with the Lie k-group structure and for which the Lie algebra is L'.

There exists a Lie group germ $G_1$ over K such that $L(G_1) = L'$ (Theorem 2). By Corollary 1 to Theorem 1 of no. 1, G and $G_1$, considered as Lie k-group germs, are locally isomorphic. Hence there exist an open neighbourhood G' of e in G and a Lie K-group germ structure on G', with Lie algebra L, which is compatible with the Lie group germ structure over k. Let V be a symmetric open neighbourhood of e in G such that $V^2 \subset G'$. Let $g \in G$. Then $\phi = \operatorname{Int} g$ is a k-isomorphism of a sufficiently small open Lie subgroup germ of G' onto an open Lie subgroup germ of G'; and $T_e(\phi)$ is K-linear and hence $T_x(\phi)$ is K-linear for x sufficiently close to e; therefore the restriction of Int g to a sufficiently small open neighbourhood of e in V is K-analytic (Differentiable and Analytic Manifolds, R, 5.14.6). By § 1, no. 9, Proposition 18, there exists on G an analytic K-manifold structure with which G is a Lie

K-group and V is an open sub-K-manifold of G. By translation, it is seen that the underlying k-manifold structure of G is the given structure. The Lie algebra of the Lie K-group G is the same as that of the open Lie K-subgroup germ V and hence is L'. Finally, the uniqueness stated in the corollary follows from § 3, no. 8, Proposition 32.