Angular momentum algebra, Wigner-Eckart theorem
With the \( 6j \) symbol defined, we can go back and and rewrite the overlap between the two ways of recoupling angular momenta in terms of the \( 6j \) symbol.
That is, we can have
$$
\vert (j_a\rightarrow [j_b\rightarrow j_c]J_{bc}) JM\rangle =\sum_{J_{ab}}(-1)^{j_a+j_b+j_c+J}\sqrt{(2J_{ab}+1)(2J_{bc}+1)}\left\{\begin{array}{ccc} j_a & j_b& J_{ab} \\ j_c & J & J_{bc} \end{array}\right\}| ([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) JM\rangle.
$$
Can you find the inverse relation?
These relations can in turn be used to write out the fully anti-symmetrized three-body wave function in a \( J \)-scheme coupled basis.
If you opt then for a specific coupling order, say \( | ([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) JM\rangle \), you need to express this representation in terms of the other coupling possibilities.