Central axis of a momentum system ($\mathbf{b}=\mathbf{P}\times \mathbf{L}/|\mathbf{P}{|}^2+\lambda \mathbf{P}$)

For nonzero total momentum $\mathbf{P}$, translating the origin by $\mathbf{b}$ changes angular momentum to $\mathbf{L}-\mathbf{b}\times \mathbf{P}$. The displayed line of origins makes this vector parallel to $\mathbf{P}$, equal to $(\mathbf{L}\cdot \mathbf{P})\mathbf{P}/|\mathbf{P}{|}^2$. The invariant $\mathbf{L}\cdot \mathbf{P}$ cannot be removed by translation; if it is zero, angular momentum vanishes on this line.