  An integral quadratic form q is usually identified with a bilinear form b
such that its Gram matrix with respect to the canonical basis is upper
triangular. Two integral quadratic forms are called strongly (resp. weakly)
Gram congruent if their corresponding upper triangular bilinear forms (resp.
their symmetrizations) are equivalent. If q is unitary, such upper triangular
bilinear form is unimodular, and one considers the associated Coxeter
transformation and its characteristic polynomial, the so-called Coxeter
polynomial of q with this identification. Two strongly Gram congruent quadratic
unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here
we show that the converse of this statement holds for the connected
non-negative case of Dynkin type A_r and arbitrary corank, and use this
characterization to complete a combinatorial classification of such quadratic
forms started in [Fundamenta Informaticae 184(1):49-82, 2021] and [Fundamenta
Informaticae 185(3):221-246, 2022].
