\item \underline{$L_zS_z$}: Acts on both and updates both spinor components
\begin{equation}
L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
\pm\frac{1}{2}m\hbar^2\ket{l,m,\pm}
\end{equation}
\item \underline{$L_zS_z$}: Acts on both and updates both spinor components
\begin{equation}
L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
\pm\frac{1}{2}m\hbar^2\ket{l,m,\pm}
\end{equation}