Rationally $4$-periodic biquotients

An $n$-dimensional manifold $M$ is said to be rationally $4$-periodic if there is an element $e\in H^4(M;\mathbb{Q})$ with the property that cupping with $e$, $\cdot \cup e:H^\ast(M;\mathbb{Q})\rightarrow H^{\ast + 4}(M;\mathbb{Q})$ is injective for $0< \ast \leq \dim M-4$ and surjective when $0\leq \ast < \dim M-4$. We classify all compact simply connected biquotients which are rationally $4$-periodic. In addition, we show that if a simply connected rationally elliptic CW-complex $X$ of dimension at least $6$ is rationally $4$-periodic, then the cohomology ring is either singly generated, or $X$ is rationally homotopy equivalent to $S^2\times \mathbb{H}P^n$, $S^3\times \mathbb{H}P^n$, or $S^3\times S^3$.