In this work, a new method to factorize certain evolution operators into an infinite product of simple evolution operators is presented. The method uses Lie operator algebra and the evolution operators are restricted to exponential form. The argument of these forms is a first-order linear partial differential operator. The method has broad applications, including the areas of sensitivity analysis, the solution of ordinary differential equations, and the solution of Liouville's equation. A sequence of xi-approximants is generated to represent the Lie operators. Under certain conditions, the covergence rate of the xi-approximant sequences is remarkably high. This work presents the general formulation of the scheme and some simple illustrative examples. Investigation of convergence properties is given in a companion paper.