# Lagrangian complement in a symplectic vector bundle

A standard, folk result in symplectic geometry states that:

in a symplectic vector bundle $(E,\pi,B,\omega)$, any lagrangian subbundle $L$ admits a lagrangian complement $L'$.

Having to use this result, without giving its proof, I would like to cite a reference which not only gives the statement but also provides a self-contained proof of it.

Would you recommend some references presenting a complete, detailed proof of the quoted result?

I do not know such a reference, but that result is a one line corollary of the existence of a compatible complex structure (on fibers). If $J(b), b\in B$ be a section of the associated bundle of compatible almost complex structures on the initial symplectic bundle then $J(b)L(b)$ be a fiber of Lagrangian complement you are looking for. –  Petya Nov 30 '13 at 18:48