he solution to this problem can be found by dealing properly with these parts of the brackets. To this end, one can consider the quantum-classical equation of motion for the density matrix $$\begin{aligned}
\frac{\partial\hat{\rho}}{\partial t}&=&
-\frac{i}{\hbar}
\left[\begin{array}{cc}\hat{H} & \hat{\rho}\end{array}\right]
\nonumber\\
&\cdot&
\left[\begin{array}{cc}
0 & 1-\frac{\hbar}{2i}
\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
%
\\
-1+\frac{\hbar}{2i}
\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
& 0\end{array}\right]
\cdot
\left[\begin{array}{c}\hat{H}\\\hat{\rho}\end{array}\right]\;.
\nonumber
\\
\label{eq:rhoW}\end{aligned}$$ As above discussed, in Eq. (\[eq:rhoW\]) the ordering problem arises from the terms in the right hand side containing the Poisson bracket operator $\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}$. Then, considering the identity $1=\hat{\rho}\cdot\hat{\rho}^{-1}=\hat{\rho}^{-1}\cdot\hat{\rho}$, Eq. (\[eq:rhoW\]) can be rewritten as $$\begin{aligned}
\partial_t \hat{\rho}
&=&
-\frac{i}{\hbar}
\left[\begin{array}{cc}\hat{H} & \hat{1}\end{array}\right]
\nonumber\\
&\cdot&
\left[\begin{array}{cc}
0 & 1-\frac{\hbar}{2i}
\left\{\ldots,\hat{\rho}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\\
-1+\frac{\hbar}{2i}
\left\{\hat{\rho},\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
& 0\end{array}\right]
\cdot
\left[\begin{array}{c}\hat{H}\\\hat{1}\end{array}\right]
\nonumber
\\
&=&
-\frac{i}{\hbar}
\left[\begin{array}{cc}\hat{H} & \hat{\rho}\hat{\rho}^{-1}\end{array}\right]
\nonumber\\
&\cdot&
\left[\begin{array}{cc}
0 & 1-\frac{\hbar}{2i}
\left\{\ldots,\hat{\rho}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\\
-1+\frac{\hbar}{2i}
\left\{\hat{\rho},\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}
& 0\end{array}\right]
\cdot
\left[\begin{array}{c}\hat{H}\\\hat{\rho}^{-1}\hat{\rho}\end{array}\right]
\nonumber
\\
%%%%
&=&
-\frac{i}{\hbar}
\left[\begin{array}{cc}\hat{H} & \hat{\rho}\end{array}\right]
\cdot
\mbox{\boldmath$\cal D$}_{\mbox{\tiny\boldmath$\cal B$},[