E0-dilation of strongly commuting CP0-semigroups

We prove that every strongly commuting pair of CP0-semigroups has a minimal E0-dilation. This is achieved in two major steps, interesting in themselves: (1) we show that a strongly commuting pair of CP0-semigroups can be represented via a two parameter product system representation; (2) we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP0-semigroups on B(H), H finite-dimensional, has an E0-dilation.