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Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant$\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preservingisometries of the five dimensional (real) hyperbolic space. We obtaindiscreteness criteria for Zariski-dense subgroups of SL(2, $\mathbb H$) usingtest maps.

Test map and Discreteness in SL(2, $\mathbb H$)