DOF) and then take its semiclassical limit [@kapra-cicco]. The resulting equation of motion is commonly known as the quantum-classical Liouville equation (QCLE) [@donoso-martens3; @donoso-martens4; @wan-schofield; @wan-schofield2; @stock; @schutte; @wan-schofield3; @schutte2; @schutte3; @mqc-rate; @schutte4; @riga-martens; @roman-martens]. The QCLE has been used to study a wide variety of problems [@thorndyke-micha; @mqc-prot; @riga-martens2; @isotope; @h-complex; @2spins; @spectra; @etransf; @opto; @prot-el; @plasmon; @prot-el2; @pump-probe; @vibr; @helix] and a number of in-depth reviews on the basic formulation of the theory exist [@rev; @simu; @sergi-theor; @rev1; @rev2; @rev3; @rev4; @rev5; @rev6; @rev7; @rev8; @rev9; @rev10; @rev11]. The mathematical structure underlying the QCLE is dictated by a quasi-Lie bracket [@b3; @b4; @sergi-spin1; @sergi-spin2]. Quasi Lie brackets are known within the community of classical molecular dynamics simulators as non-Hamiltonian brackets [@b1; @b2; @aspvg]. Mathematicians have also studied very similar structures known as almost Poisson brackets or quasi-Lie algebras [@qLie; @qLie2; @qLie3; @qLie4; @qLie5]. It is interesting to note that the quasi-Lie (or non-Hamiltonian) structure of the QCLE [@aleksandrov; @gerasimenko; @zhang-balescu; @balescu-zhang; @boucher-transchen; @gerasimenko2; @martens-fang; @prezhdo-kisil; @donoso-martens; @donoso-martens2; @qc-stat; @b3; @b4] has both favorable and unfavorable aspects associated with it. Because the antisymmetry of the quasi-Lie bracket ensures energy conservation, one is able to verify the stability of numerical 