Quaternary quadratic lattices over number fields

Abstract

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V, q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V, q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V, q).