The biquotient model of the total space of the projectivization quaternionic bundles over certain manifolds

In this paper, we compute the rational homotopy type of the quaternionic projective bundle $P(\tau ):\mathbb{H}{P}^{n-1}\to P(E)\to M$ obtain from the quaternionic tangent bundle $\tau :{\mathbb{H}}^n\to E\to M$ over quaternionic manifold $M$. If the base space $M$ is the quaternionic projective space $\mathbb{H}{P}^2$, then the rational homotopy type of the total space $P(E)$ of the quaternionic projective bundle $P(\tau )$ is given by a biquotient model $Sp(1)\mathrm{\backslash }Sp(3)\mathrm{/}Sp(1)\times Sp(1)$.