Since we have a spin orbit force which is strong, it is easy to show that the total angular momentum operator $$\hat{J}=\hat{L}+\hat{S}$$ does not commute with $\hat{L}_z$ and $\hat{S}_z$. To see this, we calculate for example $$\begin{eqnarray} [\hat{L}_z,\hat{J}^2]&=&[\hat{L}_z,(\hat{L}+\hat{S})^2] \\ \nonumber &=&[\hat{L}_z,\hat{L}^2+\hat{S}^2+2\hat{L}\hat{S}]\\ \nonumber &=& [\hat{L}_z,\hat{L}\hat{S}]=[\hat{L}_z,\hat{L}_x\hat{S}_x+\hat{L}_y\hat{S}_y+\hat{L}_z\hat{S}_z]\ne 0, \end{eqnarray}$$ since we have that $[\hat{L}_z,\hat{L}_x]=i\hbar\hat{L}_y$ and $[\hat{L}_z,\hat{L}_y]=i\hbar\hat{L}_x$.