Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form
$$\left(\begin{array}{cc} 0_r & B_1 \\ A_1 & 0_r \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{array}\right),$$
where
$$A_1 = \mathrm{I}_r \quad \text{and} \quad B_1 = \lambda \mathrm{I}_r + J_r$$
for $r$ nonzero integer and $\lambda$ nonzero complex suitable.