Exercise 3, § 10

Let G be the set of $(x, y) \in \mathbf{Q}_p \times \mathbf{Q}_p$ such that $|y| \leq 1$. This is an open subgroup of $\mathbf{Q}_p \times \mathbf{Q}_p$. Then $L(G) = \mathbf{Q}_p \times \mathbf{Q}_p$. Let $\alpha$ be the infinitesimal automorphism $(x, y) \mapsto (0, x)$. Then the conclusion of Proposition 3 is false.