Combining the last two equations from the previous slide and and applying the Wigner-Eckart theorem, we arrive at (rearranging phase factors) $$ \langle (j_aj_b)J||W^{r}||(j_cj_d)J'\rangle=\sqrt{(2J+1)(2r+1)(2J'+1)}\sum_{m_iM,M'}\left(\begin{array}{ccc} J & r & J' \\ -M & m_r & M'\end{array}\right) $$ $$ \times\left(\begin{array}{ccc} j_a &j_b & J \\ m_a &m_b &-M \end{array}\right) \left(\begin{array}{ccc} j_c &j_d &J' \\ -m_c &-m_d &M' \end{array}\right) \left(\begin{array}{ccc} p & q & r \\ -m_p&-m_q &m_r \end{array}\right) $$ $$ \times\left(\begin{array}{ccc} j_a &j_c &p \\ m_a &-m_c &-m_p \end{array}\right)\left(\begin{array}{ccc} j_b &j_d &q \\ m_b &-m_d &-m_q \end{array}\right)\langle j_a||T^p||j_c\rangle \times \langle j_b||U^q||j_d\rangle $$ which can be rewritten in terms of a \( 9j \) symbol as $$ \langle (j_aj_b)J||W^{r}||(j_cj_d)J'\rangle=\sqrt{(2J+1)(2r+1)(2J'+1)}\langle j_a||T^p||j_c\rangle \langle j_b||U^q||j_d\rangle\left\{\begin{array}{ccc} j_a & j_b& J \\ j_c & j_d & J' \\ p & q& r\end{array}\right\}. $$