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Certain vertex operator algebras have integral forms (integral spans of baseswhich are closed under the countable set of products). It is unclear when they(or integral multiples of them) are integral as lattices under the naturalbilinear form on the VOA. We show that lattice-integrality may be arrangedunder some hypotheses, including cases of integral forms invariant by finitegroups. In particular, there exists a lattice-integral Monster-invariantintegral form in the Moonshine VOA.

Lattice-integrality of certain group-invariant integral forms in vertex operator algebras