Construction algorithms for rational cubic surfaces

By a criterion of Swinnerton-Dyer it is known that a smooth cubic surface S defined over Q is birationally trivial over Q if and only if S(Q)≠∅ and S contains a Gal(Q‾,Q)-stable set of 2, 3 or 6 skew lines. In this text we describe birationally trivial smooth cubic surfaces over Q and provide algorithms to construct explicit cubic surfaces for each of the cases in Swinnerton-Dyer's criterion.