Cohomology ring of quaternionic projective space ($H^{\ast }({\mathbb{HP}}^n;\mathbb{Z})=\mathbb{Z}[u]/(u^{n+1}),|u|=4$)

The cell structure gives one integral cohomology generator in each degree divisible by four through $4n$. The Gysin sequence of a sphere bundle for the quaternionic tautological line bundle makes multiplication by its Euler class an isomorphism between successive such degrees. Choose $u$ to be the negative of that Euler class; its powers generate the displayed cohomology ring.