Reflections on S3 and Quaternionic Möbius Transformations

Let S 3 be the set of unit quaternions, let H be the algebra of quaternions, and let H * be the space of pure quaternions.It is an elementary fact that S 3 and H * {} are homeomorphic spaces by a stereographic projection.We show that a reflection in S 3 induces a linear fractional transformation on H * {} that is defined by a matrix in a symplectic group Sp(2) .In addition, we identify the left eigenvalues of such a matrix, and show the subgroup G generated by these matrices satisfies G/(I 2 ) O(4) .