This paper describes a matrix algebra realization of Clifford's theory
of biquaternions. By examining 4 x 4 skew-symmetric matrices, the pap
er shows the connection between infinitesimal screws in elliptic three
-space and vector quaternions. By studying the matrix exponential of t
he skew-symmetric matrices, the paper also shows how finite screws in
elliptic three-space lead to matrix realization of quaternions. Finall
y, 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.