Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then Ratner's orbit closure theorem states that for any unipotent subgroup $U$ of $G$, the closure $\overline{\Gamma U}$ is always of the form $\Gamma H$ for some closed subgroup $H$ of $G$ (see here. Note that Ratner's theorem holds for much more general groups than the $G$ considered here).
Now I would like to consider some particular $U$, which is a maximal abelian unipotent subgroup. Then the closure $\overline{\Gamma U}$ is of the form $\Gamma H$ for some closed subgroup $H$ of $G$.
My question is: what can we say about $H$? Is it true that $H=G$? I guess this is a rather trivial question, yet I failed to find an answer by myself nor on the internet. Any help would be appreciated!
