Equivariant vector fields on non-trivial SOn-torsors and differential Galois theory

We show how to produce SOn-equivariant vector fields on non-trivial SOn-torsors which correspond to quadratic forms non-equivalent to the unit form. For n 3 we then give an example of a Picard–Vessiot extension with group SOn which is the function field of a non-trivial SOn-torsor.