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We present a novel representation of the Lorentz group, the geometric versionof which uses "reversions" of a sphere while the algebraic version usespseudounitary 2x2 matrices over complex numbers and quaternions, and Cliffordalgebras in general. A remarkably simple formula for relativistic compositionof velocities and an accompanying geometric construction follows. The method isderived from the diffeomorphisms of the celestial sphere induced by Lorentzboost.

Cromlech, menhirs and celestial sphere: an unusual representation of the Lorentz group