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We show a Gottlieb element in the rational homotopy of a simply connectedspace $X$ implies a structural result for the Sullivan minimal model, withdifferent results depending on parity. In the even-degree case, we prove arational Gottlieb element is a terminal homotopy element. This fact allows usto complete an argument of Dupont to prove an even-degree Gottlieb elementgives a free factor in the rational cohomology of a formal space of finitetype. We apply the odd-degree result to affirm a special case of the$2N$-conjecture on Gottlieb elements of a finite complex. We combine ourresults to make a contribution to the realization problem for the classifyingspace $B\mathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying$B\mathrm{aut}_1(X_{\mathbb{Q}}) \simeq S_{\mathbb{Q}}^{2n}$ must haveinfinite-dimensional rational homotopy and vanishing rational Gottlieb elementsabove degree $2n-1$ for $n= 1, 2, 3.$

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