the quantum theory. Poisson brackets have a symplectic structure that is easily represented in matrix form [@mccauley; @goldstein]. Both Poisson brackets and commutators define Lie algebras. In terms of commutators, a Lie algebra possesses the following properties: $$\begin{aligned}
[\hat\chi_1,\hat\chi_2]&=&
-[\hat\chi_2,\hat\chi_2], \label{antisymmetry} \\
\left[\chi_1\hat\chi_2,\hat\chi_3\right]&=&
\hat\chi_1[\hat\chi_2,\hat\chi_3]+
[\hat\chi_1,\hat\chi_3]\hat\chi_2,
\\
\left[c,\hat\chi_j\right]&=& 0
\label{lie-last},\end{aligned}$$ where $c$ is a so-called c-number and $\hat \chi_j$, $j=1,2,3$ are quantum operators. In order to have a Lie algebra, together with Equations (\[antisymmetry\])–(\[lie-last\]), the Jacobi relation must also hold $${\cal J}=[\hat\chi_1,[\hat\chi_2,\hat\chi_3]]+
[\hat\chi_3,[\hat\chi_1,\hat\chi_2]]+
[\hat\chi_2,[\hat\chi_3,\hat\chi_1]]
=0 \label{jacobi}.$$

The time-translation invariance of the commutator algebra follows from the Jacobi relation, which therefore states an integrability condition. If $\hat H$ is not explicitly time-dependent, the antisymmetry of the commutator (\[antisymmetry\]), arising from the antisymmetry of the symplectic matrix $\mbox{\boldmath$\cal B$}$, ensures that the energy is a constant of motion: $d\hat H/dt= i \hat{\cal L}\hat H=0$. Energy conservation under time-translation is a fundamental property shared by the algebra of Poisson brackets and the algebra of commutators that is in agreement with Noether theorem.

Now, let us consider a hybrid quantum-classical system, in which the quantum subsystem, described by a few canonically conjugate operators $(\hat q, \hat p)=\hat x$ is embedded in a classical bath with 