On Picard–Vessiot extensions with group PGL3

Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F-irreducible PGL3-torsors in terms of nine-dimensional central simple algebras over F. We use this to construct a Picard–Vessiot extension which is the function field of a non-trivial torsor and which is a generic extension for PGL3.