We generalize an algorithm established in earlier work to compute finitelymany generators for a subgroup of finite index of a group actingdiscontinuously on hyperbolic space of dimension $2$ and $3$, to hyperbolicspace of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators up to finiteindex of a discrete subgroup of Vahlen's group, i.e. a group of $2$-by-$2$matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group rings and this newalgorithm allows to handle unit groups of orders in $2$-by-$2$ matrices overrational quaternion algebras. The rings investigated are part of the so-calledexceptional components of a rational group algebra.
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