Pure Picard-Vessiot extensions with generic properties

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