  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 $S^3\times S^3$.
