Effective structure theorems for symplectic spaces via height

Given a $2k$-dimensional symplectic space $(Z,F)$ in $N$ variables, $1 < 2k \leq N$, over a global field $K$, we prove the existence of a symplectic basis for $(Z,F)$ of bounded height. This can be viewed as a version of Siegel's lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of $(Z,F)$ into hyperbolic planes, as well as the existence of two generating flags of totally isotropic subspaces. These present analogues of known results for quadratic spaces. A distinctive feature of our argument is that it works simultaneously for essentially any field with a product formula, algebraically closed or not. In fact, we prove an even more general version of these statements, where canonical height is replaced with twisted height. All bounds on height are explicit.