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 cite and use this result,  I would like to find 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.