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 $$\left(\begin{array}{cc} 0_r & B_0 \\ A_0 & 0_s \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{array}\right),$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{s \times (s+1)} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \left(\begin{array}{cccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(r+1) \times r} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{r \times (r+1)}.$$