Cohomology ring of the orthogonal complex line flag manifold ($H^{\ast }(X_n;\mathbb{Z})=\mathbb{Z}[x,y]/(x^n,{\sum }_{j=0}^{n-1}{x}^{n-1-j}{y}^j)$)

For the orthogonal complex line flag manifold, take $x,y$ to be the first Chern classes of the duals of its two tautological lines, both in degree two. The Whitney sum formula for Chern classes gives $c_i(L^{\perp })=x^i$. The projective bundle definition of Chern classes gives the second relation, and Leray-Hirsch theorem gives the integral basis $x^i{y}^j$ with $0\le i<n$, $0\le j<n-1$. Polynomial division by the monic relation in $y$ proves that there are no further relations. Multiplying that relation by $y-x$ also gives $y^n=0$.