Components of the force and quantum numbers

The tensor force is given by $$ S_{12} (\hat r) = \frac{3}{r^2}\left(\mathbf{\sigma}_1\cdot \mathbf{r}\right) \left(\mathbf{\sigma}_2\cdot \mathbf{r}\right) -\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2$$ where the Pauli matrices are defined as $$ \sigma_x =\begin{Bmatrix} 0 & 1 \\ 1 & 0 \end{Bmatrix}, $$ $$ \sigma_y =\begin{Bmatrix} 0 & -\imath \\ \imath & 0 \end{Bmatrix}, $$ and $$ \sigma_z =\begin{Bmatrix} 1 & 0 \\ 0 & -1 \end{Bmatrix}, $$ with the properties \( \sigma = 2\mathbf{S} \) (the spin of the system, being \( 1/2 \) for nucleons), \( \sigma^2_x=\sigma^2_y=\sigma_z=\mathbf{1} \) and obeying the commutation and anti-commutation relations \( \{\sigma_x,\sigma_y\} =0 \) \( [\sigma_x,\sigma_y] =\imath\sigma_z \) etc.