Matrices as a diagonal quadratic form over rings of integers of certain quadratic number fields

Let O denote the ring of integers of a quadratic field Q(−7). In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form a1X12+a2X22+a3X32 where ai∈O for 1≤i≤3 for representing all 2×2 matrices over O. Let K denote a quadratic field such that its ring of integers OK is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that Q(−7) is the only imaginary quadratic field with the above properties. Let DK denote the discriminant of K. We have DK≡1(mod 8) if and only if 2 is a product of two distinct primes in OK. With OK as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form ∑i=1maiXi2 where ai∈OK, 1≤i≤m to represent all 2×2 matrices over OK. This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022].