Rotations and units in quaternion algebras

Unit groups of orders in quaternion algebras over number fields provide important examples of non-commutative arithmetic groups. Let be a quadratic field with d<0 a square-free integer such that , and let R be its ring of integers. In this note we study, through its representation in SO3(R), the group of units of several orders in the quaternion algebra over K with basis {1,i,j,k} satisfying the relations i2=j2=−1, ij=−ji=k.