Unimodular embeddings

In 1994 Morris Newman showed that a unimodular quadratic form on a lattice over a principal ideal domain can be represented by a triple-diagonal matrix of a rather special form, though the matrices associated with a given lattice in this way are generally not unique. The present paper considers positive definite unimodular lattices over the integers, and it begins the exploration of connections between those special matrix representations for a given lattice and the isometry class of that lattice.