I was working on a question that asks me to determine whether it is possible for a 2-dimensional subspace in a 4-dimensional symplectic vector space is neither symplectic nor Lagrangian, as well as how my conclusion can be generalised to high dimensions. Here is the work that I have done which I believe should be correct.
Claim: If a 2-dimensional subspace $W$ in a 4-dimensional symplectic vector space $V$is not symplectic, then it is Lagrangian.
Suppose $W$ is not symplectic, then let $\{ \vec {v}_1,\vec {v}_2\}$ be a basis for $W$.We have $\omega(\vec v_1,\vec v_2)=0$. Writing any $\vec w_1,\vec w_2\in W$ as $a_1\vec v_1+a_2\vec v_2$ and $b_1\vec v_1+b_2\vec v_2$ respectively. It is not hard to see that $\omega(\vec w_1,\vec w_2)=0$.
It follows that $W\subset W^{\omega}$: For any $\vec w_0\in W$, if $\vec w\in W$, then $\omega(\vec w_0,\vec w)=0$. Since $\vec w_0$ is arbitrary, we conclude that $\vec w\in W^{\omega}$. Since $\dim(W)+\dim(W^{\omega})=4$, $\dim(W)=\dim(W^{\omega})$. From $W\subset W^{\omega}$, it follows that $W=W^{\omega}$.
Let $W$ be an $n$ dimensional subspace of a $2n$ dimensional symplectic space. If $n>2$, it is possible that $W$ is neither symplectic nor Lagrangian.
If $n$ is odd, $W$ is automatically not symplectic. Let $\{\vec e_1,\vec f_1,\vec e_2,\vec f_2,\dots ,\vec e_n,\vec f_n\}$ be a symplectic basis for $V$. Let $W=$Span$\{\vec e_1,\vec e_2,\dots ,\vec e_{n-1},\vec f_{n-1}\}$. Notice that $W^{\omega}=$Span$\{\vec e_1,\vec e_2,\dots, \vec e_{n-2},\vec{e}_n,\vec{f}_n\}$, since $W^{\omega}$ is n-dimensional and for any $\vec w_1\in W$ and $\vec w_2\in W^{\omega}$, $\omega(\vec w_1,\vec w_2)=0$. Hence, $W\neq W^{\omega}$. If $n$ is even, then construct $W$ similarly, and notice that while $\vec e_1\neq 0$, $\omega(\vec e_1,w)=0$ for any $w\in W$. Hence, $W$ is not symplectic. Showing that $W$ is not Lagrangian is similar.
My question is that under what conditions is $W$ not symplectic and not Lagrangian when $n>2$. Using the definition for symplectic space to construct a basis for $W$ seems too complicated, so is there a simpler way of figuring out the conditions?