We present a new method for constructing simple ordinary abelian surfaces with a small embedding degree. To a quartic CM field
, we associate a quadric surface
(ℚ) and use its parametrization to determine Weil numbers in
corresponding in the sense of Honda-Tate theory to such surfaces. In general, the resulting surfaces have parameter
≈ 8. However, if there exist rational lines on
, they can be used to achieve
≈ 4. We give examples of non-primitive quartic CM fields such that
has rulings by rational lines. Furthermore, we show how our method can be used to construct parametric families of pairing-friendly surfaces.