Let
$$B = \frac{1}{4} \left(\begin{array}{cccc} 0 & -5 & 0 & -3 \\ 5 & 0 & 3 & 0 \\ 0 & -3 & 0 & -5 \\ 3 & 0 & 5 & 0 \end{array}\right).$$
Determine a real number $a$ and a matrix $P$ such that
$$P \in \mathcal{O}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R}) \quad \text{and} \quad P^{\top} B P = \left(\begin{array}{cccc} 0 & a & 0 & 0 \\ -a & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/a \\ 0 & 0 & -1/a & 0 \end{array}\right).$$