If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a Coxeter element in the Weyl group?
I know that the Bruhat cells that are not the big cell are closed so they are not containing any open sets, but maybe the union of all the conjugation in $G(\mathbb Z)$ of Bruhat-Coxeter cells is Zariski dense in $G(\mathbb C)$?

Edit:$H$ is also of finite index.