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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?

As usual, any comments about how improve this question are welcome.

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    $\begingroup$ 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. $\endgroup$
    – Petya
    Commented Nov 30, 2013 at 18:48
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    $\begingroup$ In "Lectures on Symplectic Manifolds" by Weinstein that is written on the page 9. $\endgroup$
    – Petya
    Commented Nov 30, 2013 at 19:19

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