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Many thanks for the idea, I tried to realize it (determinant 1 is OK, but we want it to be equivalent over ${\mathbb Q}$ with a unimodular lattice). I guess this is just my ignorance, could you point me to appropriate reference?
I still don't quite understand how it can be embedded to a given lattice (non-unimodular). For unimodular it's fine, except that I still cannot find a proper reference for an embedding result even in this generality (Morrison quotes Nikulin, but this result is not easy to find in Nikulin's paper).
