We note that, using the anti-commuting properties of the creation operators, we obtain $$N_{ab}\sum_{m_am_b}\langle j_am_aj_bm_b|JM\rangle \vert\Phi^{ab}\rangle=-N_{ab}\sum_{m_am_b}\langle j_am_aj_bm_b|JM\rangle\vert\Phi^{ba}\rangle.$$ Furthermore, using the property of the Clebsch-Gordan coefficient $$\langle j_am_aj_bm_b|JM>=(-1)^{j_a+j_b-J}\langle j_bm_bj_am_a|JM\rangle,$$ which can be used to show that $$|(j_bj_a)JM\rangle = \left\{a^{\dagger}_ba^{\dagger}_a\right\}^J_M|^{16}\mathrm{O}\rangle=N_{ab}\sum_{m_am_b}\langle j_bm_bj_am_a|JM\rangle|\Phi^{ba}\rangle,$$ is equal to $$|(j_bj_a)JM\rangle=(-1)^{j_a+j_b-J+1}|(j_aj_b)JM\rangle.$$