Oscillator bilinear commutator
= Oscillator bilinear commutator
For bosonic modes satisfying $[a_i,a_j^\dagger]=\delta_{ij}$, the number-preserving bilinears $T_{ij}=a_i^\dagger a_j$ obey
$$
[T_{ij},T_{kl}]=\delta_{jk}T_{il}-\delta_{il}T_{kj}.
$$
= Oscillator bilinear commutator
For bosonic modes satisfying $[a_i,a_j^\dagger]=\delta_{ij}$, the number-preserving bilinears $T_{ij}=a_i^\dagger a_j$ obey
$$
[T_{ij},T_{kl}]=\delta_{jk}T_{il}-\delta_{il}T_{kj}.
$$