# Commutation of bosonic operators on finite Hilbert space

I understand the common commutation relations for creationd and annihilation operators, given by: $$[b_i, b_j^\dagger]|n_1 n_2\dots n_N\rangle = b_ib_j^\dagger|n_1 n_2\dots n_N\rangle-b_j^\dagger b_i|n_1 n_2\dots n_N\rangle\\ = \sqrt{n_i(n_j+1)}|n_1\dots n_i-1\ n_j+1\dots n_N\rangle\\ -\sqrt{n_i(n_j+1)}|n_1\dots n_i-1\ n_j+1\dots n_N\rangle=0$$ My question is about equation (2.57) on page 32. They restrict the Fock space to contain at most $$N_P$$ bosons per site, and then they affirm that the following relations hold: $$\left[\bar{b}_{i}, \bar{b}_{j}\right]=0,\left[\bar{b}_{i}, \bar{b}_{j}^{\dagger}\right]=\delta_{i j}\left[1-\frac{N_{P}+1}{N_{P} !}\left(\bar{b}_{i}^{\dagger}\right)^{N_{P}}\left(\bar{b}_{i}\right)^{N_{P}}\right]$$ Where does that come from? I started considering the general Fock state $$|n_1 n_2\dots n_N\rangle=\prod_{k=1}^{N}\frac{1}{\sqrt{n_k!}}{b_k^\dagger}^{n_k}|0\rangle$$ and trying to compute the commutator expression but obtained nothing useful. Can anyone help me?