and Machida [@Ohmi] and use a Cartesian basis ($\psi =\{ \psi_{x}, \psi_{y}, \psi_{z}\}$). This representation is related to the spherical ($\{1, 0, -1\}$) basis as follows [@mueller]: $\psi_{x} = \frac{1}{\sqrt{2}}(\psi_{1} - \psi_{-1})$,$\psi_{x} = \frac{i}{\sqrt{2}}(\psi_{1} + \psi_{-1})$ and $\psi_{z} = \psi_{0}$.

In the Cartesian basis the irreducible decomposition of a spin-$1$ system is $\textbf{A} = A_{0}\textbf{I} + i \epsilon_{abc}A^{(1)}_{c} + A^{(s)}$, where the scalar $A_{0} = \frac{\text{Tr}(A)}{3}$, $\textbf{A}^{(a)} = \epsilon_{abc}A^{(1)}_{c}$ is a completely antisymmetric matrix proportional to the vector spin $\langle\textbf{S}\rangle$ order, and $\textbf{A}^{(s)}$ is a symmetric traceless tensor which is related to the spin fluctuations $\langle \textbf{S}_{a}\textbf{S}_{b} + \textbf{S}_{b}\textbf{S}_{a} \rangle$, and is a nematic degree of freedom.

Writing the density matrix ($\textbf{F}$) and interaction ($\textbf{V}$) in terms of their respective irreducible decompositions we obtain three equations of motion: $\frac{\partial}{\partial t}F_{0} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{0} = 0$, $\frac{\partial}{\partial t}{F^{(a)}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F^{(a)} = i[\textbf{V}^{(a)}, \textbf{F}^{(a)}] + i[\textbf{V}^{(s)}, \textbf{F}^{(s)}]$, and $\frac{\partial}{\partial t}{F^{(s)}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F^{(s)} = i[\textbf{V}^{(a)}, \textbf{F}^{(s)}] + i[\textbf{V}^{(s)}, \textbf{F}^{(a)}]$. Here the second equation describes the evolution of $\langle \textbf{S} \rangle$ and the spin currents, while the last equation describes dynamics of the nematicity ($\textbf{A}^{(s)}$).

The new feature is that the spin and nematic degrees of freedom are coupled by the interaction matrix. Using the 