  It is known that the groups of Euclidean rotations in dimension 3 (isometries
of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic
isometries in dimension 3), and screw motions in dimension 3 can be represented
by the groups of unit--norm elements in the algebras of real quaternions,
biquaternions (complex quaternions) and dual quaternions, respectively. In this
work, we present a unified framework that allows a wider scope on the subject
and includes all the classical results related to the action in dimension 3 and
4 of unit--norm elements of the algebras described above and the algebra of
split biquaternions as particular cases. We establish a decomposition of
unit--norm elements in all cases and obtain as a byproduct a new decomposition
of the group rotations in dimension 4.
