= Solution
For $T_{ij}=a_i^\dagger a_j$, expand the commutator and commute one annihilation operator past one creation operator:
$$
\begin{aligned}
[T_{ij},T_{kl}]
&=a_i^\dagger a_j a_k^\dagger a_l
-a_k^\dagger a_l a_i^\dagger a_j\\
&=\delta_{jk}a_i^\dagger a_l
-\delta_{il}a_k^\dagger a_j.
\end{aligned}
$$
Hence the <oscillator bilinear commutator> is
$$
\boxed{[T_{ij},T_{kl}]=\delta_{jk}T_{il}-\delta_{il}T_{kj}}.
$$
Since
$$
H=T_{xx}+T_{yy}+1,
$$
we obtain
$$
[T_{ij},H]
=\sum_k(\delta_{jk}T_{ik}-\delta_{ik}T_{kj})=0.
$$
Every $T_{ij}$ therefore acts within a fixed energy eigenspace.
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