Separated Lie models and the homotopy Lie algebra

A simply connected topological space XX has homotopy Lie algebra π∗(ΩX)⊗Qπ∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of XX, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.