Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group G G over an algebraically closed field C C of characteristic 0, we construct a pure Picard-Vessiot extension for G G , namely, a Picard-Vessiot extension E ⊃ F \mathcal E\supset \mathcal F , with differential Galois group G G , such that E \mathcal E and F \mathcal F are purely differentially transcendental over C C . The differential field E \mathcal E is the quotient field of a G G -stable proper differential subring R \mathcal R with the property that if F F is any differential field with field of constants C C and E ⊃ F E\supset F is a Picard-Vessiot extension with differential Galois group a connected subgroup H H of G G , then there is a differential homomorphism ϕ : R → E \phi :\mathcal R\rightarrow E such that E E is generated over F F as a differential field by ϕ ( R ) \phi (\mathcal R) .