Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.

Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of finite index according to [this MO question](http://mathoverflow.net/questions/133072/on-the-f-rational-points-of-the-derived-group-of-a-connected-reductive-algebra).

My question is : do we have (maybe under stronger assumptions) $U(F) \subset (G(F),G(F))$ for any unipotent subgroup $U \subset G$ ?

If no, same question for a given unipotent subgroup $U \subset G$.