What question did this study set out to answer?

The aim is to investigate how reflections in the unit quaternion set S 3 create linear transformations in pure quaternions.

March 29, 2026Open Access

Reflections on S3 and Quaternionic Möbius Transformations

Key Points

The aim is to investigate how reflections in the unit quaternion set S 3 create linear transformations in pure quaternions.
Defined the relationship between unit quaternions and pure quaternions using stereographic projection.
Analyzed reflections in S 3 and their effects on linear fractional transformations in H *.
Identified left eigenvalues of matrices in the symplectic group Sp(2) and their subgroup properties.
Demonstrated that reflections in S 3 induce specific linear fractional transformations in H *.
Identified distinct left eigenvalues corresponding to matrices in Sp(2).
Established that the subgroup G generated by these matrices satisfies G/(I 2 ) isomorphic to O(4).

Abstract

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).

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