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.