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

We study subgroups generated by prescribed unipotent elements. For $n\ge 3$, let $$\mathrm{\Gamma }(Q)=⟨E_{ij}(q_{ij}):i\mathrm{\ne }j⟩$$ be the subgroup of $\mathrm{SL}(n,\mathbb{R})$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $\mathrm{\Gamma }(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective: it determines the relevant ring of $S$-integers, a diagonal conjugating matrix, and an explicit description of the resulting subgroup by congruence conditions. We then study the rank-one family $${\mathrm{\Gamma }}_q=⟨\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 0\\ q& 1\end{array}\right)⟩,q=\frac{s}{t}\in \mathbb{Q}\mathrm{.}$$ For $q\mathrm{\ne }0,\pm 3$, we prove that $${\mathrm{\Gamma }}_q={\mathrm{\Gamma }}_1^{(t)}(s)$$ if and only if its upper-triangular subgroup strictly contains $$⟨\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)⟩\mathrm{.}$$ Thus the congruence-subgroup problem is reduced to constructing a single upper-triangular element outside this cyclic subgroup. As applications, we reinterpret several constructions from the study of non-freeness as constructions of arithmetic groups. We verify the criterion for all rational parameters $q=\frac{s}{t}\in (-4,4)$ with $1\le \mathrm{\mid }s\mathrm{\mid }\le 21$, and obtain new infinite families of congruence subgroups from indefinite binary quadratic forms and Pell-type equations.