integration algorithms. However, because the quasi-Lie algebra is not invariant under time translation, the initially classical DOF acquire a quantum character as time flows, implying that one never has a true dynamical theory of quantum and classical DOF but only an approximated dynamics of a full quantum system [@caro-salcedo]. This is somewhat paradoxical because energy conservation is linked to time-translation symmetry through the Noether theorem; nevertheless, quasi-Lie brackets break the time-translation symmetry of the algebra (which can be seen as a signature of the effect of the classical bath on the quantum subsystem).

This review deals with situations where the bath DOF are described in terms of non-canonical coordinates [@non-c; @mccauley] or non-Hamiltonian coordinates [@b1; @b2; @aspvg], and situations where dissipation must be taken explicitly into account [@ray-barbados; @langevin-qc]. In all these cases, we will see that the operator-valued probability functions will develop new functional dependences and novel definitions of the quasi-Lie brackets will have to be introduced. In particular, we will first describe the case of a classical spin bath [@sergi-spin1; @sergi-spin2], as an example of a bath described by non-canonical coordinates [@non-c; @mccauley]. It has been shown that for such a bath an off-diagonal [@manini] open-path [@pati; @filipp; 