+To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+\begin{eqnarray}
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
+&=&
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+\end{eqnarray}
+All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$.