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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.
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