The traditional approach towards generating lift-and-project cuts involves solving a cut generating linear program (CGLP) that grows prohibitively large if a multiple-term disjunction is used instead of the classical dichotomy. In this paper we present an alternative approach towards generating lift-and-project cuts that solves the cut generation problem in the original space (i.e. does not introduce any extra variables). We present computational results that show our method to be superior to the classical approach. As soon as the number of terms in the disjunction grows larger than 4, the benefit of not solving the large CGLP directly is clear, and the gap grows larger as more terms are included in the disjunction.