2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224288We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will show that the birational map is a blow up followed by a blow down. This will allow us to use the blow up formula for motives together with Vishik's work on the motives of quadrics to give a motivic decomposition of X(J).20 pages, 1 figureAlgebraic GeometryRings and Algebras11E04 (Primary) 14E05, 14L30, 14C15 (Secondary)text