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Do these Zariski-dense subgroups of $\operatorname{SO}_{6}(\mathbb C)$ have non-empty inters...

Let $G\leq \operatorname{SO}_{6}(\mathbb Z)$ be a finite-index normal subgroup, so it's a Zariski dense subgroup of $\operatorname{SO}_{6}(\mathbb C)$; and let $H$ be the subset of $\operatorname{SO}_ …