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. We describe the general situation, the case where
G is a spin group and V is a half-spin representation, and recent work with Anne
     Queguiner-Mathieu that generalizes Pfister's theorem in dimension 12.