A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over the ring of integers of a global field, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property discovered by V. Arnold for the case of integral binary quadratic forms.
Earnest, A. G. and Fitzgerald, Robert W. "Represented Value Sets for Integral Binary Quadratic Forms and Lattices." (Jan 2007).