Spin-one spectrum of a quantum asymmetric top ($E={\hslash }^2\{a_x+a_y,a_y+a_z,a_z+a_x\}$)

Within the $l=1$ multiplet, $|1,0⟩$ has energy ${\hslash }^2(a_x+a_y)$. The symmetric and antisymmetric combinations of $|1,1⟩$ and $|1,-1⟩$ have the other two pairwise coefficient sums. Principal-axis permutation relates these energies; equality requires corresponding coefficient degeneracies rather than angular-momentum conservation alone.