If I am not mistaken, the characteristic polynomial of a matrix in $\mathfrak{s}\mathfrak{p}_4(F)$ is of the form $T^4+aT^2+b$ for some $a,b$. In particular, $0$ always has even multiplicity as an eigenvalue of a matrix in $\mathfrak{s}\mathfrak{p}_4(F)$. Do you mean with "coming in pairs" that if $\lambda$ is an eigenvalue so is $-\lambda$? I will have a look later at the reference you gave.

Just the algebraically closed case would be enough :), including both the zero and prime characteristic case if possible.