Our two-body state can thus be written as $$ |(j_1j_2)JM_J\rangle=\sum_{m_{j_1}m_{j_2}}\langle j_1m_{j_1}j_2m_{j_2}|JM_J\rangle|j_1m_{j_1}\rangle|j_2m_{j_2}\rangle. $$ Due to the coupling order of the Clebsch-Gordan coefficient it reads as \( j_1 \) coupled to \( j_2 \) to yield a final angular momentum \( J \). If we invert the order of coupling we would have $$ |(j_2j_1)JM_J\rangle=\sum_{m_{j_1}m_{j_2}}\langle j_2m_{j_2}j_1m_{j_1}|JM_J\rangle|j_1m_{j_1}\rangle|j_2m_{j_2}\rangle, $$ and due to the symmetry properties of the Clebsch-Gordan coefficient we have $$ |(j_2j_1)JM_J\rangle=(-1)^{j_1+j_2-J}\sum_{m_{j_1}m_{j_2}}\langle j_1m_{j_1}j_2m_{j_2}|JM_J\rangle|j_1m_{j_1}\rangle|j_2m_{j_2}\rangle=(-1)^{j_1+j_2-J}|(j_1j_2)JM_J\rangle. $$ We call the basis \( |(j_1j_2)JM_J\rangle \) for the coupled basis, or just \( j \)-coupled basis/scheme. The basis formed by the simple product of single-particle eigenstates \( |j_1m_{j_1}\rangle|j_2m_{j_2}\rangle \) is called the uncoupled-basis, or just the \( m \)-scheme basis.