For a semisimple algebraic group G acting on a vector space V, it is rare that there is a dense G-orbit in the projective space P(V). When there is such an orbit, one can reduce the study of G-torsors to that of torsors under a smaller group H. This provides a unified view of several classical algebraic results, such as the diagonalization of symmetric matrices and Pfister’s theorems on quadratic forms of dimension ¡= 12 in I^{3}.W edescribethegeneralsituation, thecasewhereGisaspingroupandV isahalf− spinrepresentation, andrecentworkwithAnneQueguiner−MathieuthatgeneralizesP fister′stheoremindimension12.