Angular momentum algebra, Wigner-Eckart theorem
We use then the latter equation to define the so-called \( 6j \)-symbols
$$
\begin{eqnarray}
\langle (j_a\rightarrow [j_b\rightarrow j_c]J_{bc}) J'M'| ([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) JM\rangle & = & \delta_{JJ'}\delta_{MM'}\sum_{m_am_bm_c}\langle j_am_aj_bm_b|J_{ab}M_{ab}\rangle \langle J_{ab}M_{ab}j_cm_c|JM\rangle \\ \nonumber
& & \times \langle j_bm_bj_cm_c|J_{bc}M_{bc}\rangle \langle j_am_aJ_{bc}M_{bc}|JM\rangle \\ \nonumber
& =&(-1)^{j_a+j_b+j_c+J}\sqrt{(2J_{ab}+1)(2J_{bc}+1)}\left\{\begin{array}{ccc} j_a & j_b& J_{ab} \\ j_c & J & J_{bc} \end{array}\right\} ,
\end{eqnarray}
$$
where the symbol in curly brackets is the \( 6j \) symbol.
A specific coupling order has to be respected in the symbol, that is, the so-called triangular relations between three angular momenta needs to be respected, that is
$$
\left\{\begin{array}{ccc} x & x& x \\ & & \end{array}\right\}\hspace{0.1cm}\left\{\begin{array}{ccc} & & x \\ x& x & \end{array}\right\}\hspace{0.1cm}\left\{\begin{array}{ccc} & x& \\ x & &x \end{array}\right\}\hspace{0.1cm}\left\{\begin{array}{ccc} x & & \\ & x &x \end{array}\right\}\hspace{0.1cm}
$$