Using this relation we get $$ \begin{eqnarray*} \left[{\bf r} \otimes {\bf r} \right]^{(2)}_\mu &=& \sqrt{4\pi}r^2 \sum_{lm} \sum_{\alpha ,\beta} \langle 1\alpha 1\beta\vert 2\mu \rangle \\ &&\times \langle 1\alpha 1\beta\vert l-m \rangle \frac{(-1)^{1-1-m}}{\sqrt{2l+1}} \left(\begin{array}{ccc} 1&1&l \\ 0 &0 &0\end{array}\right)Y_{l-m}(-1)^m\\ &=& \sqrt{4\pi}r^2 \left(\begin{array}{ccc} 1&1&2 \\ 0 &0 &0\end{array}\right) Y_{2-\mu}\\ &=& \sqrt{4\pi}r^2 \sqrt{\frac{2}{15}}Y_{2-\mu} \end{eqnarray*} $$