Corollary 2

Let $G$ be a Lie group, $H$ an integral subgroup of $G$ and $a \in L(G)$. Consider the following conditions: (i) $a \in L(H)$; (ii) $\exp_G(\lambda a) \in H$ for all $\lambda \in K$. Then (i) $\Rightarrow$ (ii). If the topology of $H$ admits a countable base, then (i) $\Leftrightarrow$ (ii). Let $i$ be the canonical injection of $H$ into $G$. If $a \in L(H)$, then

Hence (i) $\Rightarrow$ (ii). The converse when $H$ has a countable base follows from Proposition 3.