 Solution

About a fixed origin, the total momentum and angular momentum areA time-independent translation ${\mathbf{x}}_i^{\prime }={\mathbf{x}}_i-\mathbf{b}$ leaves every velocity unchanged. ThereforeThe cross product $\mathbf{b}\times \mathbf{P}$ is perpendicular to $\mathbf{P}$, giving the invariantThe other requested transformation follows from the vector triple-product identity:The actual PDF asks about $\mathbf{L}\times \mathbf{P}$; the TeX's occurrence of $\mathbf{F}$ is a transcription error.

If $\mathbf{P}=0$, the translation term vanishes and angular momentum is independent of origin. If $\mathbf{P}\ne 0$, chooseThen $\mathbf{b}\times \mathbf{P}=\mathbf{L}-(\mathbf{L}\cdot \mathbf{P})\mathbf{P}/|\mathbf{P}{|}^2$, and soThus the new angular momentum is parallel to the total momentum. The freedom along $\mathbf{P}$ gives the central axis of a momentum system. If the invariant dot product is zero, the new angular momentum is the zero vector. All expressions assume a translated fixed origin, not a moving reference frame.