  Herein we shall consider Lorentz boosts and Wigner rotations from a
(complexified) quaternionic point of view. We shall demonstrate that for a
suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz
boosts can be phrased in terms of the quaternion square root of the relative
4-velocity connecting the two inertial frames. Straightforward computations
then lead to quite explicit and relatively simple algebraic formulae for the
composition of 4-velocities and the Wigner angle. We subsequently relate the
Wigner rotation to the generic non-associativity of the composition of three
4-velocities, and develop a necessary and sufficient condition for
associativity to hold. Finally, we relate the composition of 4-velocities to a
specific implementation of the Baker-Campbell-Hausdorff theorem. As compared to
ordinary 4x4 Lorentz transformations, the use of self-adjoint complexified
quaternions leads, from a computational view, to storage savings and more rapid
computations, and from a pedagogical view to to relatively simple and explicit
formulae.
