  We present a novel representation of the Lorentz group, the geometric version
of which uses "reversions" of a sphere while the algebraic version uses
pseudounitary 2x2 matrices over complex numbers and quaternions, and Clifford
algebras in general. A remarkably simple formula for relativistic composition
of velocities and an accompanying geometric construction follows. The method is
derived from the diffeomorphisms of the celestial sphere induced by Lorentz
boost.
