Torque About the Center of Mass: Orbital and Spin Equations

Write $\vec{F}^{\text{ext}} = \frac{d}{dt}(m_T\vec{V}_{\text{cm}})$ and state the model behind it: the whole system as one point particle of mass $m_T$ at the center of mass, with all external forces acting there.

Substitute $\vec{r}_{S,i} = \vec{r}_{S,\text{cm}} + \vec{r}_{\text{cm},i}$ into the torque sum to get $\vec{\tau}_S = \vec{r}_{S,\text{cm}} \times \vec{F}^{\text{ext}} + \vec{\tau}_{\text{cm}}$, write the matching orbital-plus-spin split of $\vec{L}_S$, and note which term is independent of $S$.

Differentiate $\vec{L}_S^{\text{orbital}}$ and $\vec{L}_{\text{cm}}^{\text{spin}}$ with the cross-product rule, show the parallel-vector terms and the $\vec{A}_{\text{cm}}$ term vanish, and arrive at $\vec{\tau}_{S,\text{cm}} = d\vec{L}_S^{\text{orbital}}/dt$ and $\vec{\tau}_{\text{cm}} = d\vec{L}_{\text{cm}}^{\text{spin}}/dt$.