 Solution

The real symmetric quadratic form is positive for every nonzero vector of occupied-virtual rotation parameters precisely when its matrix $X$ is positive definite. Then the Hartree-Fock determinant is a strict local minimum within the considered real orbital-rotation space. A negative eigenvalue gives an unstable lowering direction; a zero eigenvalue requires higher-order analysis. Stability in this space does not by itself prove global minimality or stability under more general spin/complex-orbital variations.