Gottlieb elements and rational homotopy types realized as classifying spaces

In this paper, we are interested in the realizability problem as classifying spaces in rational homotopy theory. Namely, if a given space Y can appear as the classifying space $B{\mathrm{aut}}_1\left(X\right)$ up to rational homotopy for some space X. We prove the non-realization of $S^{2n}$ for $n\le 5$ under the hypothesis of X having non-vanishing Gottlieb elements above degree $2n-1$, which extends the results of Lupton and Smith. Moreover, we also show some realization and non-realization results for products of Eilenberg-Mac Lane spaces under the hypothesis of X having finitely many non-zero homotopy groups. Our proofs are based on a technique of constructing specific derivations of a Sullivan minimal algebra with a given Gottlieb element.