I found the operators to be $J_z, J^2, L^2, S^2$, but how do I prove that they commute?
My attempt:
For $L^2$, we know that $[\vec{L},L^2]=0$, so $[\vec{L} \cdot \vec{S}, L^2]=0$. But I don't understand why.
For $S^2$, we know that $[\vec{S},S^2]=0$, so $[\vec{L} \cdot \vec{S}, S^2]=0$. But I'm not sure why.
For $J^2=L^2+S^2+2\vec{L} \cdot \vec{S}$. We also know that $[\vec{L} \cdot \vec{S},\vec{S} \cdot \vec{L}] = 0$, so $[\vec{L} \cdot \vec{S}, J^2]=0$
For $J_z$ ???