Representation of the direct sum of two quadratic fields by rational symmetric matrices

Let $f$ be a fourth-degree polynomial over the field of rational numbers $\mathbf Q$ with leading coefficient $I$ which decomposes over $\mathbf Q$ into the product of two irreducible second-degree polynomials. It is proved that in order that $f$ be the characteristic polynomial of a symmetric matrix with elements in $\mathbf Q$, it is necessary and sufficient that all the roots of $f$ be real.