Groups Generated by Root Unipotents: Higher-rank and rank-one

For $n\geq 3$, let \[ Γ(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle \] be the subgroup of $\operatorname{SL}(n,\mathbb R)$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $Γ(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitra