I am reading the following paper about multi-mode Floquet theory. They have the following matrix $H_F$ given with entries
$({H_F})_{nk} = (E+n\omega) \delta_{n,k}+\sum_{i=1}^2 V^i \left( \delta_{n-k,N_i}+\delta_{n-k,-N_i}\right)$
where $n,k$ go from $-\infty$ to $\infty$ and $N_i$ are integers. $\omega$ is a frequency an thus $\omega \in \mathbb{R}$. In addition there exist parameters $\omega_1$ and $\omega_2$ such that $\omega_1 = N_1 \omega$ and $\omega_2 = N_2 \omega$. The matrix is an infinitely dimensional matrix with one diagonal and four off-diagonals. This matrix is the standard Floquet parametrization.
Now, the paper claims that this matrix can be block-diagonalized into blocks $(H_F)^p$ which are characterized by an integer $p$ which is either zero or which can't be written as $p=n_1N_1+n_2N_2$ with $n_1,n_2 \in \mathbb{Z}$. But, let us just focus on the case, where $N_1$ and $N_2$ are chosen such that every $p\in \mathbb{Z}$ can be written by the sum $n_1N_1+n_2N_2$. Then, only one block, namely $(H_F)^0$ exists which is given by
$H^{0}_{n_1n_2k_1k_2} = (E+n_1\omega_1+n_2\omega_2)\delta_{n_1,k_1}\delta_{n_2,k_2}+\sum_{i=1}^2 V^i \left(\delta_{n_i-k_i,1}+\delta_{n_i-k_i,-1} \right)$
which should be unterstood as an infinitely sized matrix $(H^0)_{n_2k_2}$ with each entry being an infinitely size matrix in indices $(n_1,k_1)$. Thus one can map $n_1,n_2 \to n$ and $k_1,k_2 \to k$ to have a "normal" matrix parametrization.
I am majorly lost on how to get from the first expression to the second. The block-diagonalized form having four parameters completely throws me off.
I would be thankful for any advice. I couldn't really find the keyword to find useful literature to explain this.