Cohomology ring of the connected sum of two complex projective planes ($\mathbb{Q}[a_2,b_2]/(ab,a^2-b^2)$)

For the standard complex orientations, the two summand classes have zero product and equal squares, the oriented degree-four class. The quadratic relations imply all cubic monomials vanish. The rational Betti numbers are $1,2,1$ in degrees zero, two and four.