= Two-transitive binary quadratic-form actions
{title2=$Sp_{2m}(2)\curvearrowright Q^\pm(B)$}
The <symplectic group over a finite field> is transitive on each of the two quadratic types. The stabilizer of a refinement is its <orthogonal group over a finite field>. In the displayed parametrization, other refinements of the same type correspond to nonzero singular vectors, and refinements of opposite type to nonsingular vectors. <Witt's lemma> makes the orthogonal stabilizer transitive on each of these vector sets. Thus both type actions are <two-transitive> for $m\ge2$, and a minus stabilizer is transitive on the plus orbit. For $m=1$ the minus orbit is a singleton.
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