On the nucleus of the Grassmmann embedding of the symplectic dual polar spaces DSp(2n,F), char(F)=2

Let $n \geq 3$ and let $F$ be a field of characteristic 2. Let $DSp(2n,F)$ denote the dual polar space associated with the building of Type $C_n$ over $F$ and let $\mathcal{G}_{n-2}$ denote the $(n-2)$-Grassmannian of type $C_n$. Using the bijective correspondence between the points of $\mathcal{G}_{n-2}$ and the quads of $DSp(2n,F)$, we construct a full projective embedding of $\mathcal{G}_{n-2}$ into the nucleus of the Grassmann embedding of $DSp(2n,F)$. This generalizes a result of the paper \cite{CaLu2007} which contains an alternative proof of this fact in the case when $n=3$ and $F$ is finite.