Angular momentum algebra, two-body matrix elements

The two-body matrix element is a scalar and since it obeys rotational symmetry, it is diagonal in \( J \), meaning that the corresponding matrix element in \( J \)-scheme is $$ \langle (j_aj_b) JM | \hat{V} | (j_cj_d) JM \rangle = N_{ab}N_{cd}\sum_{m_am_bm_cm_d}\langle j_am_aj_bm_b|JM\rangle $$ $$ \times \langle j_cm_cj_dm_d|JM\rangle\langle (j_am_aj_bm_b)M | \hat{V} | (j_cm_cj_dm_d)M \rangle, $$ and note that of the four \( m \)-values in the above sum, only three are independent due to the constraint \( m_a+m_b=M=m_c+m_d \).