We have thus the coupled basis $$|(j_1j_2)JM_J\rangle=\sum_{m_{j_1}m_{j_2}}\langle j_1m_{j_1}j_2m_{j_2}|JM_J\rangle|j_1m_{j_1}\rangle|j_2m_{j_2}\rangle.$$ and the uncoupled basis $$|j_1m_{j_1}\rangle|j_2m_{j_2}\rangle.$$ The latter can easily be generalized to many single-particle states whereas the first needs specific coupling coefficients and definitions of coupling orders. The $m$-scheme basis is easy to implement numerically and is used in most standard shell-model codes. Our coupled basis obeys also the following relations $$\hat{J}^2|(j_1j_2)JM_J\rangle=\hbar^2J(J+1)|(j_1j_2)JM_J\rangle$$ $$\hat{J}_z|(j_1j_2)JM_J\rangle=\hbar M_J|(j_1j_2)JM_J\rangle,$$