Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let Cox(Sr) be the homogeneous coordinate ring of the blow-up Sr of P2 in r general points, i.e., a smooth Del Pezzo surface of degree 9−r. We prove that for r∈{6,7}, Proj(Cox(Sr)) can be embedded into Gr/Pr, where Gr is an algebraic group with root system given by the primitive Picard lattice of Sr and Pr⊂Gr is a certain maximal parabolic subgroup.