Cohomology automorphisms of a product of two complex projective spaces
= Cohomology automorphisms of a product of two complex projective spaces
For $n\geq1$, every graded-ring automorphism of
$$
H^*(\mathbb{CP}^n\times\mathbb{CP}^n;\mathbb Z)
\cong\mathbb Z[x,y]/(x^{n+1},y^{n+1})
$$
acts on $H^2$ by a signed permutation matrix. All eight such matrices are realized by independently applying complex conjugation to the factors and by interchanging the factors.