We want to rewrite $$ \langle \Phi_i \vert O_{\mu}^{\lambda} \vert \Phi_j\rangle, $$ in terms of the reduced matrix element only. Let us introduce the relevant quantum numbers for the states \( \Phi_i \) and \( \Phi_j \). We include only the relevant ones. We have then in \( m \)-scheme $$ \langle \Phi_i \vert O_{\mu}^{\lambda} \vert \Phi_j\rangle = \sum_{pq} \langle p \vert O_{\mu}^{\lambda} \vert q \rangle \langle \Phi_{M}^{J} \vert a^{\dagger}_pa_q \vert \Phi_{M'}^{J'}\rangle. $$ With a shell-model \( m \)-scheme basis it is straightforward to compute these amplitudes. However, as mentioned above, if we wish to related these elements to experiment, we need to use the Wigner-Eckart theorem and express the amplitudes in terms of reduced matrix elements.