Proposition 2

Let G be a Lie group over K, k a non-discrete closed subfield of K and H a Lie subgroup of the Lie k-group G. Suppose that $L(H)$ is a vector sub-K-space of $L(G)$ which admits a topological supplement. Then H is a Lie subgroup of the Lie K-group G.

There exists a Lie subgroup germ $H'$ of the Lie K-group G such that $L(H') = L(H)$ (Theorem 3). Consider G, H, $H'$ as Lie k-group germs; Theorem 3 then proves that $H \cap H'$ is open in H and $H'$. Hence there exists an open neighbourhood U of e in G such that $U \cap H$ is a submanifold of G over K. Therefore, H is a Lie subgroup of the Lie K-group G (§ 1, no. 3, Proposition 6).