\textbf{Two linear maps: nilpotent case}\\
We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. In this question, we assume that $M$ is nilpotent.

Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form
$$\begin{pmatrix} 0_r & B_0 \\ A_0 & 0_s \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{pmatrix},$$
where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs:
$$A_0 = \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{pmatrix}_{s \times (s+1)} \quad \text{and} \quad B_0 = \begin{pmatrix} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{pmatrix}_{(s+1) \times s};$$
$$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$
$$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$
$$A_0 = \begin{pmatrix} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{pmatrix}_{(r+1) \times r} \quad \text{and} \quad B_0 = \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{pmatrix}_{r \times (r+1)}.$$