Angular momentum algebra, Wigner-Eckart theorem

We wish to apply the above definitions to the computations of a matrix element $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle, $$ where we have skipped a reference to specific single-particle states. This is the expectation value for two specific states, labelled by angular momenta \( J' \) and \( J \). These states form an orthonormal basis. Using the properties of the Clebsch-Gordan coefficients we can write $$ T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle=\sum_{J''M''}\langle \lambda \mu J'M'|J''M''\rangle|\Psi^{J''}_{M''}\rangle, $$ and assuming that states with different \( J \) and \( M \) are orthonormal we arrive at $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle= \langle \lambda \mu J'M'|JM\rangle \langle \Phi^J_M|\Psi^{J}_{M}\rangle. $$