$4 \times 4 $ Matrix Representations of Hurwitz Split Quaternions

In this paper, we investigate the algebraic properties of Hurwitz split quaternions through their matrix representations. We construct the $4 \times 4$ 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.