You might want to study the work of Detinko, Flannery, and co-authors. For example:

<cite authors="Detinko, A.; Flannery, D. L.; Hulpke, A.">_Detinko, A.; Flannery, D. L.; Hulpke, A._, [**Zariski density and computing in arithmetic groups**](http://dx.doi.org/10.1090/mcom/3236),  [ZBL06825254](https://zbmath.org/?q=an:06825254).</cite>

<cite authors="Detinko, A. S.; Flannery, D. L.; Hulpke, A.">_Detinko, A. S.; Flannery, D. L.; Hulpke, A._, [**Algorithms for arithmetic groups with the congruence subgroup property.**](http://dx.doi.org/10.1016/j.jalgebra.2014.08.027), J. Algebra 421, 234-259 (2015). [ZBL1319.20040](https://zbmath.org/?q=an:1319.20040).</cite>