  Quaternions have an (over a century-old) extensive and quite complicated
interaction with special relativity. Since quaternions are intrinsically
4-dimensional, and do such a good job of handling 3-dimensional rotations, the
hope has always been that the use of quaternions would simplify some of the
algebra of the Lorentz transformations. Herein we report a relatively nice
result for the relativistic combination of non-collinear 3-velocities. If we
work with the relativistic half-velocities $w$ defined by $v={2w\over1+w^2}$,
and promote them to quaternions using $\mathbf{w} = w \; \mathbf{\hat n}$,
where $\mathbf{\hat n}$ is a unit quaternion, then we shall show \[
\mathbf{w}_{1\oplus2} = \mathbf{w}_1 \oplus \mathbf{w}_2
=(1-\mathbf{w}_1\mathbf{w}_2)^{-1} (\mathbf{w}_1 +\mathbf{w}_2) = (\mathbf{w}_1
+\mathbf{w}_2)(1-\mathbf{w}_2\mathbf{w}_1)^{-1}. \] All of the complicated
angular dependence for relativistic combination of non-collinear 3-velocities
is now encoded in the quaternion multiplication of $\mathbf{w}_1$ with
$\mathbf{w}_2$. This result can furthermore be extended to obtain an elegant
and compact formula for the associated Wigner angle: \[
\mathrm{e}^{\mathbf{\Omega}} = \mathrm{e}^{\Omega \; \mathbf{\hat\Omega} } =
(1-\mathbf{w}_1\mathbf{w}_2)^{-1} (1-\mathbf{w}_2\mathbf{w}_1), \] in terms of
which \[ {\mathbf{\hat{n}}}_{1\oplus2} = \mathrm{e}^{\mathbf{\Omega}/2} \;\;
{\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}; \qquad\qquad
{\mathbf{\hat{n}}}_{2\oplus1} = \mathrm{e}^{-\mathbf{\Omega}/2} \;\;
{\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}. \] Thus, we would
argue, many key results that are ultimately due to the non-commutativity of
non-collinear boosts can be easily rephrased in terms of the algebra of
quaternions.
