Sums of squares of integral multiples of an integral element of real bi-quadratic fields

For any given positive integer m we construct certain totally positive algebraic integers \(\alpha \) $\alpha$ of a real bi-quadratic field K and obtain some necessary conditions for which \(m\alpha \) $m\alpha$ cannot be represented as sum of integral squares. We show this for integers that lie in quadratic subfields of K and for integers which are in K but not in any quadratic subfield of K. We provide examples in tabular form for each case to corroborate the results.