= Solution
{c}
From part (a), $[N,a_j]=-a_j$ and $[N,a_j^*]=a_j^*$. The <commutator product rule> gives
$$
\begin{aligned}
[N,a_1a_2^*]&=[N,a_1]a_2^*+a_1[N,a_2^*]=-a_1a_2^*+a_1a_2^*=0,\\
[N,a_2a_1^*]&=[N,a_2]a_1^*+a_2[N,a_1^*]=-a_2a_1^*+a_2a_1^*=0.
\end{aligned}
$$
Consequently
$$
\boxed{[N,L]=0.}
$$
Each term in $L$ transfers one excitation between the two modes, preserving the total <number operator> <eigenvalue>. This is an instance of the <oscillator bilinear commutator>; every <number operator> <eigenspace> is an <invariant subspace> of $L$.
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