Let $G$ be an algebraic group defined over a char 0 local field $k$. Following Borel and Tits (73) we define the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$ generated by the unipotent elements of $G(k)$.

Suppose $G$ is gegerated by a finite set of unipotent $k$-sugroups, say $U_1,\cdots, U_n$. Is it true that the group generated by $U_1(k), \cdots, U_n(k)$ is $G^+$?

I feel the answer is positive but do not know how to prove it. It seems that the ideas of the original paper of Borel and Tits can help, but I still do not read French (which I always plan to learn) yet.