The structuring effect of a Gottlieb element on the Sullivan minimal model of a space

We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the rational cohomology of a formal space of finite type. We apply the odd-degree result to affirm a special case of the $2N$-conjecture on Gottlieb elements of a finite complex. We combine our results to make a contribution to the realization problem for the classifying space $B{\mathrm{a}\mathrm{u}\mathrm{t}}_1(X)$. We prove a simply connected space $X$ satisfying $B{\mathrm{a}\mathrm{u}\mathrm{t}}_1(X_{\mathbb{Q}})\simeq S_{\mathbb{Q}}^{2n}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n=1,2,3.$