\hat W(X,t) \end{array}\right]
\mbox{\boldmath$\cal D$}
\left[\begin{array}{c} \hat H_{\rm W} \\ \hat W(X,t) \end{array}\right]
=-\frac{i}{\hbar}\left[\hat H_{\rm W}
,\hat W(X,T)\right]_{\mbox{\boldmath$\cal D$}}
=-i\hat{\cal L}^{\mbox{\boldmath$\cal D$}}\hat\chi\;,
\label{eq:qclm}$$ where $\mbox{\boldmath$\cal D$}$ is an antisymmetric matrix super-operator defined by $$\mbox{\boldmath$\cal D$}=\left[\begin{array}{cc} 0& 1-
\frac{\overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}}
{2i\hbar^{-1}}\\
-\left(1-
\frac{\overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}}
{2i\hbar^{-1}}\right) & 0\end{array}\right]\;,
\label{eq:D_matrix}$$ with $\mbox{\boldmath$\nabla$}=(\partial/\partial Q,\partial/\partial P)
=\partial/\partial X$, and $$\overleftarrow{\mbox{\boldmath$\nabla$}}\mbox{\boldmath${\cal B}$}\overrightarrow{\mbox{\boldmath$\nabla$}}
=\sum_{I,J=1}^{2N}
\overleftarrow{\nabla}_I
{\cal B}_{IJ}\overrightarrow{\nabla}_J
\label{eq:symplectic-bracket}$$ denotes the Poisson bracket operator. The last equality on the right-hand side of Equation (\[eq:qclm\]) defines the quantum-classical Liouville operator $i\hat{\cal L}^{\mbox{\boldmath$\cal D$}}$. Equation (\[eq:qclm\]) is the QCLE [@donoso-martens3; @donoso-martens4; @wan-schofield; @wan-schofield2; @stock; @schutte; @wan-schofield3; @schutte2; @schutte3; @mqc-rate; @schutte4; @riga-martens; @roman-martens] of the system.

The QCLE in Equation (\[eq:qclm\]) is founded upon a quasi-Lie bracket, which we may write explicitly as $$[\hat\chi_1(X),\hat\chi_2(X)]_{\mbox{\boldmath$\cal D$}}
=
\left[\begin{array}{cc}\hat\chi_1(X) & \hat\chi_2(X)\end{array}\right]
\mbox{\boldmath$\cal D$}
\left[\begin{array}{c}\hat\chi_1(X) \\ \hat\chi_2(X)\end{array}\right]\;,
\label{eq:gen-quantum-algebra}$$ where $\mbox{\boldmath$\cal D$}$ is the antisymmetric matrix operator defined in Equation (\[eq:D\_matrix\]). However, in contrast to the Lie brackets of quantum and classical mechanics, the quasi-Lie bracket defined in Equation (\[eq:gen-quantum-algebra\]) violates the Jacobi relation (\[jacobi\]): $$\small
{\cal J}_{\mbox{\boldmath$\cal D$}}
=\left[\hat\chi_1(X),\left[\hat\chi_2(X),\hat\chi_3(X)
\right]_{\mbox{\boldmath$\cal D$}}
\right]_{\mbox{\boldmath$\cal D$}}
+\left[\hat\chi_3(X),\left[\hat\chi_1(X),\hat\chi_2(X)
\right]_{\mbox{\boldmath$\cal D$}}
\right]_{\mbox{\boldmath$\cal D$}}
+\left[\hat\chi_2(X),\left[\hat\chi_3(X),\hat\chi_1(X)
\right]_{\mbox{\boldmath$\cal D$}}
\right]_{\mbox{\boldmath$\cal D$}} \neq 0
\;.\label{qc-jacobi}$$

The failure of the 