In this post: Passage in proving lagrangian subspaces have lagrangian complements
where the poster wants to prove that Lagrangian subspace has Lagrangian complement, one of the intermediate steps is to show that given $L$ Lagrangian and $M$ isotropic with $ M\cap L=\{0\}$, there exists $e\in M^\sigma\smallsetminus(L+M)$ where $M^\sigma$ is the orthogonal complement of $M$. I have been trying to show such an element exist but couldn't do it. How do we show this?
If $M$ is actually Lagrangian, then you are done: $M$ is a Lagrangian complement to $L$.
So assume that $M$ is isotropic but not Lagrangian, and that $M\cap L=\{0\}$. Taking symplectic orthogonals in the last equality and using that $L^{\sigma}=L$, we get $$ M^{\sigma}+L=E. $$ If $M^{\sigma}$ would be contained in $L+M$, then the above equality would imply that $L+M=E$. But along with $M\cap L=\{0\}$, this means that $M$ is actually Lagrangian, being isotropic of dimension $\dim E-\dim L$.
This is a contradiction, hence $M^{\sigma}$ is not contained in $L+M$. So there exists $e\in M^{\sigma}\setminus(L+M)$.
