  Continuum or discrete polarizable models for the study of optoelectronic
processes in embedded subsystems rely mostly on the restriction of the
surrounding electronic dielectric response to its low frequency limit. Such a
description hinges on the assumption that the electrons in the surrounding
medium react instantaneously to any excitation in the central subsystem,
treating thus the environment in the adiabatic limit. Exploiting a recently
developed embedded $GW$ formalism, with an environment described at the fully
ab initio level, we assess the merits of the adiabatic limit with respect to an
environment where the full dynamics of the dielectric response is considered.
Further, we show how to properly take the static limit of the environment
susceptibility, introducing the so-called Coulomb-hole and screened-exchange
contributions to the reaction field. As a first application, we consider a
C$_{60}$ molecule at the surface of a C$_{60}$ crystal, namely a case where the
dynamics of the embedded and embedding subsystems are similar. The common
adiabatic assumption, when properly treated, generates errors below $10\%$ on
the polarization energy associated with frontier energy levels and associated
energy gaps. Finally, we consider a water molecule inside a metallic nanotube,
the worst case for the environment adiabatic limit. The error on the gap
polarization energy remains below $10\%$, even though the error on the frontier
orbitals polarization energies can reach a few tenths of an electronvolt.
