 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

To prove existence, compare the standard isotropic flag $F_{\bullet }$ with $g{F}_{\bullet }$. The rankstogether 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 $g\in B\dot{w}B$. 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.