Weak regularity and one-parameter subgroups of definable $p$-adic Lie groups

We prove that every group $G$ definable in the pure $p$-adic field $\mathbb{Q}_p$ is weakly regular. We show that every one-parameter subgroup is definable. We also show that its one-parameter core $G_u$, together with a suitable regular open subgroup $G_\Omega$, is definable. Finally, we show that if $H$ is a dfg component of $G$, then $G_u=\langle(H^g)_u:g\in G\rangle$. In fact, $G_u$ is a finite product of the one-parameter cores of finitely many conjugates of $H$. In particular, when $G$ is definably amenable, $G_u=H_u$.