Cyclotomic matrices over real quadratic integer rings

We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings, R. Moreover, we enumerate a related class of symmetric R-matrices; those R-matrices whose eigenvalues are contained inside the interval [−2, 2] but whose characteristic polynomials are not in Z[x].