  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 $B\mathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying
$B\mathrm{aut}_1(X_{\mathbb{Q}}) \simeq S_{\mathbb{Q}}^{2n}$ must have
infinite-dimensional rational homotopy and vanishing rational Gottlieb elements
above degree $2n-1$ for $n= 1, 2, 3.$

Algebraic Topology