The quadratic sum problem for symplectic pairs

Let $(b,u)$ be a pair consisting of a symplectic form $b$ on a finite-dimensional vector space $V$ over a field $\mathbb{F}$, and of a $b$-alternating endomorphism $u$ of $V$ (i.e. $b(x,u(x))=0$ for all $x$ in $V$). Let $p$ and $q$ be arbitrary polynomials of degree $2$ with coefficients in $\mathbb{F}$. We characterize, in terms of the invariant factors of $u$, the condition that $u$ splits into $u_1+u_2$ for some pair $(u_1,u_2)$ of $b$-alternating endomorphisms such that $p(u_1)=q(u_2)=0$.