We show that a Gottlieb element in the rational homotopy of a simply connected space X induces a basis change for the Sullivan minimal model of X with structuring results depending on parity. We apply these results to give some progress on three open problems in rational homotopy. We complete an argument of Dupont to prove that an even-degree Gottlieb element in the rational homotopy of a formal space corresponds to a free factor in cohomology. We prove a special case of the 2N-conjecture concerning the location of odd-degree Gottlieb elements in a hyperbolic space. Finally, we combine our results to address the realization problem for classifying spaces Baut(X) giving constraints on a hypothetical space X such that Baut(X) is rationally an even-dimensional sphere of low degree. This is joint work with Greg Lupton.