the spin components is written as $\nabla_a^S=\partial/\partial S_a$, with $a=x,y,z$. The equations of motion of the spin are then written as $$\dot {\bf S}= {\cal B}^{\bf S} \nabla^{\bf S} H_{\rm S}\;,
\label{eq:eqofm}$$ where $$\mbox{\boldmath $\cal B$}^{\bf S}
=\left[\begin{array}{ccc} 0   &  S_z & -S_y \\ 
                         -S_z &  0   &  S_x \\
                          S_y & -S_x &  0 \end{array}\right] 
\;.
\label{eq:bmat}$$

One can also adopt the compact form ${\cal B}_{ab}^{\bf S}=\sum_{c=x,y,z}\epsilon_{abc}S_c$ and $a,b=x,y,z$ of the antisymmetric matrix $\mbox{\boldmath $\cal B$}^{\bf S}$, where $\epsilon_{abc}$ is the Levi–Civita pseudo-tensor. The Casimir $C_2={\bf S}\cdot {\bf S}$ is preserved by the equations of motion (\[eq:eqofm\]), independently of the form of the spin Hamiltonian $H^{\rm S}({\bf S})$. In addition, the dynamics has a zero phase space compressibility $\kappa^{\rm S}=\nabla{\bf S}\cdot \dot {\bf S} =0$. The classical phase space flow of the spin is defined through the non-canonical bracket $$\sum_{a,b} A({\bf S})\overleftarrow{\nabla}^{\bf S}_a{\cal B}_{ab}^{\rm S}
\overrightarrow{\nabla}^{\bf S}_b B({\bf S})
=
A({\bf S})\overleftarrow{\nabla}^{\bf S} \mbox{\boldmath$\cal B$}^{\bf S}
\overrightarrow{\nabla}^{\bf S}B({\bf S})\;,
\label{eq:ncan-brack}$$ where $A=A({\bf S})$ and $B=B({\bf S})$ are arbitrary functions of the spin DOF.

Consider now the hybrid quantum-classical Hamiltonian of a quantum subsystem coupled to the classical spin $$\begin{aligned}
\hat{\cal H}({\bf S})&=&\hat{H}(\{\hat{\chi}\})
+V_{\rm C}(\{\hat{\chi}\},{\bf S}) +H^{\bf S}({\bf S})
\nonumber\\
