While we present the idea of including a classical environment polarizability in the calculation of the screened Coulomb interaction of subsystem I after the GW-BSE-in-DFT embedding, these methods have been developed historically in different order. For the sake of presentation, we pick up from the definition of the modified Coulomb interaction in Eq. ( ) in GW-BSE-in-DFT embedding, which now in real-space form including the explicit dependence on the frequency ω reads