I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985).
Their construction goes as follows. Take a complex vector space $V$ of dimension $6$. Let $G$ be the subvariety of $\mathbb P(\Lambda^2 V)$ formed by tensors of rang $2$, and $\Delta^*$ the subvariety of $\mathbb P(\Lambda^2V^*)$ formed by tensors of rang $4$. Now choose $L \subset \mathbb P(\Lambda^2V)$ a sufficiently general linear subspace of dimension $8$ and set $S = G\cap L$, $X = \Delta^* \cap L^\perp$. It turns out that $S$ is a K3 surface in $L$ of degree 14, while $X$ is a cubic hypersurface in $L^\perp \cong \mathbb P^5$.
Now, it is a theorem of Beauville-Donagi that if $F$ is the variety of lines contained in $X$, then $F$ is isomorphic to $S^{[2]}$. In particular, this implies that there exists a canonical isomorphism $$ H^2(F,\mathbb Z) = H^2(S,\mathbb Z)\oplus \mathbb Z \delta,$$ where $2\,\delta$ is the class of the exceptional divisor in $S^{[2]}$.
Let $l$ be the hyperplane class in $H^2(S,\mathbb Z)$ given by the embedding in $L$ and $g$ be the hyperplane class of $F$ given by the Plücker embedding.
Why does the following relation hold? $$g = 2\,l - 5\,\delta.$$
