Corollary 2

Let $G$ be a finite-dimensional Lie group over $K$, $H$ a subgroup of $G$, $V$ an open neighbourhood of $e$ in $G$ and $(M_i)_{i \in I}$ a family of analytic manifolds over $K$; for all $i \in I$, let $f_i$ be a $K$-analytic mapping of $V$ into $M_i$ such that

(i) If $K = \mathbf{C}$, $H$ is a Lie subgroup of $G$. (ii) If $K$ is a finite extension of $\mathbf{Q}_p$ and $I$ is finite, $H$ is a Lie subgroup of $G$.

(i) Suppose that $K = \mathbf{C}$. We consider $G$ as a real Lie group. Then $H$ is a real Lie subgroup of $G$ (Theorem 2). Let $a \in L(H)$. There exists a connected open neighbourhood $W$ of $0$ in $\mathbf{C}$ such that $\exp \lambda a \in V$ for all $\lambda \in W$. Let $i \in I$. Then $f_i(\exp \lambda a) = f_i(e)$ if $\lambda \in \mathbf{R} \cap W$. Hence $f_i(\exp \lambda a) = f_i(e)$ if $\lambda \in W$ by analytic continuation. Thus, $\exp \lambda a \in H$ for $\lambda \in W$ and therefore $\mu a \in L(H)$ for all $\mu \in \mathbf{C}$. Therefore $H$ is a Lie subgroup of the complex Lie group $G$ (§ 4, no. 2, Proposition 2).

(ii) Suppose that $K$ is a finite extension of $\mathbf{Q}_p$. We consider $G$ as a Lie group over $\mathbf{Q}_p$. It is finite-dimensional and Theorem 2 implies that $H$ is a $p$-adic Lie subgroup of $G$. Since $I$ is finite, $\prod_{i \in I} M_i$ is a manifold and it can be assumed that the family $(f_i)$ reduces to a single mapping $f$. Let $a \in L(G)$. Let $\phi$ be an exponential mapping of $G$. Then $f(\phi(\lambda a)) = f(e)$ for $\lambda \in \mathbf{Q}_p$ and $|\lambda|$ sufficiently small. Since $f$ is K-analytic, it follows that $f(\phi(\lambda a)) = f(e)$ for $\lambda \in K$ and $|\lambda|$ sufficiently small. Hence $\phi(\lambda a) \in H$ for $\lambda \in K$ and $|\lambda|$ sufficiently small and therefore $\mu a \in L(H)$ for all $\mu \in K$. The proof is completed as in (i).

Corollary 2 (ii) is no longer true if we omit the hypothesis that I is finite.