= Lagrangian complement
{c}
{title2=$E=L\oplus K$}
A <Lagrangian complement> to $L$ in a <symplectic vector space> is a <Lagrangian subspace> $K$ with $E=L\oplus K$. Such complements exist smoothly for Lagrangian subbundles. Identify any initial complement with $L^*$ using the symplectic cross-pairing. If its internal pairing is $b(\eta,\zeta)$, correct its lift by a vector $a\eta\in L$ with $\zeta(a\eta)=-b(\eta,\zeta)/2$. The corrected complement has zero internal pairing.
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