Question 705027

The total angular momentum $\vec{J} = \vec{L} + \vec{S} = \vec{L} + \frac{\hbar}{2}\vec{\sigma}$. The spin-orbit coupling term is proportional to $\vec{L} \cdot \vec{S}$. Since the total angular momentum commutes with the spin-orbit interaction, $\vec{L} + \frac{\hbar}{2}\vec{\sigma}$ commutes with $\vec{\sigma} \cdot \vec{L}$.