Starting with $$ \langle (j_aj_b)J||W^{r}||(j_cj_d)J'\rangle\equiv\sum_{M,m_r,M'}(-1)^{J-M}\left(\begin{array}{ccc} J & r & J' \\ -M & m_r & M'\end{array}\right) $$ $$ \times\langle (j_aj_bJM|\left[ T^{p}_{m_p}U^{q}_{m_q} \right]^{r}_{m_r}|(j_cj_d)J'M'\rangle, $$ we assume now that \( T \) acts only on \( j_a \) and \( j_c \) and that \( U \) acts only on \( j_b \) and \( j_d \). The matrix element \( \langle (j_aj_bJM|\left[ T^{p}_{m_p}U^{q}_{m_q} \right]^{r}_{m_r}|(j_cj_d)J'M'\rangle \) can be written out, when we insert a complete set of states \( |j_im_ij_jm_j\rangle\langle j_im_ij_jm_j| \) between \( T \) and \( U \) as $$ \langle (j_aj_bJM|\left[ T^{p}_{m_p}U^{q}_{m_q} \right]^{r}_{m_r}|(j_cj_d)J'M'\rangle=\sum_{m_i}\langle pm_pqm_q|rm_r\rangle\langle j_am_aj_bm_b|JM\rangle\langle j_cm_cj_dm_d|J'M'\rangle $$ $$ \times \langle (j_am_aj_bm_b|\left[ T^{p}_{m_p}\right]^{r}_{m_r}|(j_cm_cj_bm_b)\rangle\langle (j_cm_cj_bm_b|\left[ U^{q}_{m_q}\right]^{r}_{m_r}|(j_cm_cj_dm_d)\rangle. $$ The complete set of states that was inserted between \( T \) and \( U \) reduces to \( |j_cm_cj_bm_b\rangle\langle j_cm_cj_bm_b| \) due to orthogonality of the states.