Using the reduced matrix element of the spin operators defined as $$ \begin{eqnarray*} \langle S\vert \vert\left[{\bf \sigma}_1 \otimes {\bf \sigma}_2\right]^{(2)} \vert \vert S' \rangle & = & \sqrt{(2S+1)(2S'+1)5} \left\{\begin{array}{ccc} s_1&s_2&S \\s_3&s_4&S' \\ 1&1&2\end{array}\right\}\\ &\times& \langle s_1 \vert \vert {\bf \sigma}_1 \vert \vert s_3\rangle \langle s_2 \vert\vert {\bf \sigma}_2 \vert \vert s_4\rangle \end{eqnarray*} $$ and inserting these expressions for the two reduced matrix elements we get $$ \begin{array}{ll} &\\ \langle lSJ\vert V\vert l'S'J\rangle =&(-1)^{S+J}\sqrt{30(2l+1)(2l'+1)(2S+1)(2S'+1)}\\ &\times\left\{\begin{array}{ccc}l&S &J \\l'&S&2\end{array}\right\} \left(\begin{array}{ccc}l&2&l'\\0&0&0\end{array}\right) \left\{\begin{array}{ccc}s_{1}&s_{2}&S\\s_{3}&s_{4}&S'\\ 1&1&2\end{array} \right\}\\ &\times\langle s_{1}\vert\vert \sigma_{1}\vert\vert s_{3}\rangle \langle s_{2}\vert\vert \sigma_{2}\vert \vert s_{4}\rangle. \end{array} $$