Jacobi implies that the algebra of quasi-Lie brackets is not invariant under time-translation. For example, it can be generally proven that $$\begin{aligned}
e^{it{\cal L}^{\mbox{\boldmath$\cal D$}}}\left[\hat\chi_1(X,0),
\hat\chi_2(X)\right]
\neq
\left[e^{it{\cal L}^{\mbox{\boldmath$\cal D$}}}\hat\chi_1(X),
e^{it{\cal L}^{\mbox{\boldmath$\cal D$}}}\hat\chi_2(X)\right] \;.\end{aligned}$$ On the other hand, the quasi-Lie bracket conserves the energy $e^{it{\cal L}^{\mbox{\boldmath$\cal D$}}}\hat H {_{\rm W}}(X)
= \hat H{_{\rm W}}(X)$. [Hence, the dynamics generated by the QCLE displays energy conservation and lack of time-translation invariance of the bracket algebra. The situation is surprising because one does not expect a broken time-translation invariance symmetry in an isolated system. However, while a total hybrid quantum-classical system is closed from the point of view of energy conservation, the quasi-Lie bracket describes the irreversible transfer of quantum information from the subsystem to the classical DOF, which acquire a quantum character as the time flows. In this sense,]{} one can heuristically argue that the lack of time-translation invariance or the algebra is a mere consequence of the open dynamics of the quantum subsystem.

Derivation of the QCLE through a Partial Wigner Transform
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When the bath DOF are described by canonically conjugate variables (and only in this case), the hybrid quantum can be derived by performing a partial Wigner transform of 