On the matrix realization of the theory of biquaternions

This paper describes a matrix algebra realization of Clifford’s theory of biquaternions. By examining 4 × 4 skew-symmetric matrices, the paper shows the connection between infinitesimal screws in elliptic three-space and vector quaternions. By studying the matrix exponential of the skew-symmetric matrices, the paper also shows how finite screws in elliptic three-space lead to matrix realization of quaternions. Finally, it is shown that line transformations in elliptic three-space lead to double quaternions and that a dual quaternion is a limiting case of a double quaternion.