Let $\mathbf{v}=(V_1,V_2,V_3)$, $\mathbf{x}=(X_1,X_2,X_3)$ and $\{\sigma_x,\sigma_y,\sigma_z\}$ be the Pauli matrices. I am trying to understand the physical meaning of the geometric product of the cross-product with itself, and specifically its invariance group:
$$ \begin{align} f(\mathbf{v},\mathbf{x})&=(\mathbf{x}\times\mathbf{v})^2\\ &=(\sigma_x(V_2X_3-V_3X_2)+ \sigma_y(V_1X_3-V_3X_1)+\sigma_z(V_1X_2-V_2X_1))^2\\ &=(V_1X_2-V_2X_1)^2+(V_1X_3-V_3X_1)^2+(V_2X_3-V_3X_2)^2 \end{align} $$
The function can also be written as follows
$$ f(\mathbf{v},\mathbf{x})=\left(\det \pmatrix{\sigma_x&\sigma_y &\sigma_z\\X_1&X_2&X_3\\V_1&V_2&V_2} \right)^2 $$
I am trying to understand if it has any connection to group theory.