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April 1, 2026Open Access

4 4 Matrix Representations of Hurwitz Split Quaternions

Key Points

The research aims to explore the algebraic properties of Hurwitz split quaternions using their matrix representations.
Constructed $4 \times 4$ left and right matrix representations.
Analyzed the block structure of these matrices.
Established the left representation as a homomorphism and the right as an anti-homomorphism.
Investigated properties of the matrices, including trace and determinant.
Proved that the trace of the matrices is always an even integer.
Showed that the determinant corresponds to the square of the split quaternion norm.

Abstract

In this paper, we investigate the algebraic properties of Hurwitz split quaternions through their matrix representations. We construct the 4 4 left and right matrix representations and demonstrate that they have a specific block structure. Furthermore, we establish that the left representation is a homomorphism, while the right representation is an anti-homomorphism. Finally, we investigate certain properties of these matrices, proving that the trace is always an even integer and the determinant corresponds to the square of the split quaternion norm.

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