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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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