  We consider three families of groups: the Bianchi groups SL(2,O) where O is
the ring of integers of an imaginary, quadratic field; the groups SL*(2,O)
where O is a *-order of a definite, rational quaternion algebra with an
orthogonal involution; and the groups SL(2,O) where O is an order of a
definite, rational quaternion algebra. We show that such groups are generated
by elementary matrices if and only if O is semi-Euclidean (or
*-semi-Euclidean), which is a generalization of the usual notion of a Euclidean
ring. The proofs are surprisingly simple and proceed by considering fundamental
domains of Kleinian groups.
