  We generalize an algorithm established in earlier work \cite{algebrapaper} to
compute finitely many generators for a subgroup of finite index of an
arithmetic group acting properly discontinuously on hyperbolic space of
dimension $2$ and $3$, to hyperbolic space of higher dimensions using Clifford
algebras.
  We hence get an algorithm which gives a finite set of generators of finite
index subgroups 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 new
algorithm allows to handle unit groups of orders in $2$-by-$2$ matrices over
rational quaternion algebras. The rings investigated are part of the so-called
exceptional components of a rational group algebra.
