 Angular momentum decomposition about the centre of mass

Write each position and velocity as its centre of mass value plus a relative value. Expanding the total angular momentum gives the displayed split into orbital angular momentum of the centre and internal angular momentum about it. The cross terms vanish because ${\sum }_i{m}_i{\mathbf{r}}_i^{\prime }={\sum }_i{m}_i{\mathbf{v}}_i^{\prime }=0$.