The universal fibration with fibre X in rational homotopy theory

Let X be a simply connected space with finite-dimensional rational homotopy groups. Let p(infinity):UE -> Baut(1)(X) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map omega:aut(1)(Baut(1)(X-Q))-> Baut(1)((X)Q) expressed in terms of derivations of the relative Sullivan model of p(infinity). We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space Baut(1)(X-Q) as a consequence. We also prove that CPQn cannot be realized as Baut(1)(X-Q) for n <= 4 and X with finite-dimensional rational homotopy groups.