A note on the codimension of the linear section of the Lagrangian-Grassmannian L(6,12)

Consider a $2n$-dimensional symplectic vector space $E$ over an arbitrary field $\mathbb{F}$. Given a contraction map $f: \wedge^n E \rightarrow \wedge^{n-2} E$suchthattheLagrangian--Grassmannian$L(n,2n)=G(n,2n)\cap{\mathbb P}(\ker f)$,where$\wedge^r E$denotesthe$r$-th exterior power of $E$ and $\mathbb{P}(\ker f)$ is the projectivization of $\ker f$. In this paper, for a symplectic vector space $E$ of dimension $n=6$, we prove that the surjectivity of the contraction map $f:\wedge^{6} E \rightarrow \wedge^{4} E$ depends on the characteristic of the base field and we calculate the codimension of the linear section ${\mathbb P}(\ker f)\subseteq {\mathbb P}(\wedge^{6}E)$ for any characteristic.