Angular momentum algebra, Wigner-Eckart theorem
However, when we deal the same state in an angular momentum coupled basis, we need to be a little bit more careful. We can namely couple the states
as follows
$$
\vert([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) J\rangle= \sum_{m_am_bm_c}\langle j_am_aj_bm_b|J_{ab}M_{ab}\rangle \langle J_{ab}M_{ab}j_cm_c|JM\rangle|j_am_a\rangle\otimes |j_bm_b\rangle \otimes |j_cm_c\rangle \ ,
\tag{1}
$$
that is, we couple first \( j_a \) to \( j_b \) to yield an intermediate angular momentum \( J_{ab} \), then to \( j_c \) yielding the final angular momentum \( J \).