Boson-pair Casimir operator ($K_3^2-K_1^2-K_2^2=[(N_a-N_b{)}^2-1]/4$)

The identities $K_{+}{K}_{-}=N_a{N}_b$ and $K_{-}{K}_{+}=(N_a+1)(N_b+1)$ give the displayed quadratic Casimir operator. It commutes with all generators because the number difference does. A fixed number-difference sector therefore has a fixed Casimir value even though pair creation changes its total occupation.