A scalar expression for matrices with symplectic involution

Abstract:

Various algebraic reductions are made to facilitate computer verification of the following result: If x and y are $8 \times 8$ matrices such that [x, y] is regular, $\operatorname {tr} (x) = 0$ , and, with respect to the canonical symplectic involution, x is symmetric and y is antisymmetric, then the element ${(x + [x,y]x{[x,y]^{ - 1}})^2}$ satisfies a minimal equation of degree $\leqslant 2$.