Sum of Squares over Rationals

Recently it has been shown that a multivariate (homogeneous) polynomial with rational coefficients that can be written as a sum of squares of forms with real coefficients, is not necessarily a sum of squares of forms with rational coefficients. Essentially, only one construction for such forms is known, namely taking the $K/\Q$-norm of a sufficiently general form with coefficients in a number field $K$. Whether this construction yields a form with the desired properties depends on Galois-theoretic properties of $K$ that are not yet well understood. We construct new families of examples, and we shed new light on some well-known open questions.