= Solution
For each signed permutation $w\in W$, choose its symplectic signed-permutation matrix $\dot w\in N_G(T)$. The <Bruhat decomposition of a reductive algebraic group> is the explicit disjoint union
$$
\operatorname{Sp}_{2n}
=\bigsqcup_{w\in(\mathbb Z/2)^n\rtimes S_n}B\dot wB.
$$
To prove existence, compare the standard isotropic flag $F_\bullet$ with $gF_\bullet$. The ranks
$$
r_{ij}=\dim(F_i\cap gF_j)
$$
together with the symplectic orthogonality relations determine a unique signed permutation $w$. Symplectic row and column operations from $B$ then reduce $g$ to $\dot w$, so $g\in B\dot wB$. Conversely, the same intersection dimensions are constant on a double coset and recover $w$, proving disjointness. This is symplectic Gaussian elimination and establishes the claimed decomposition.
Solved by gpt-5.6-sol high.
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