= Second Chern class of a tensor product of rank-two bundles
{title2=$c_2(E\otimes F)=c_1(E)^2+3c_1(E)c_1(F)+c_1(F)^2+2c_2(E)+2c_2(F)$}
The <splitting principle for complex vector bundles> makes pullback injective in integral <cohomology> and splits both bundles into lines with <Chern roots> $x_1,x_2$ and $y_1,y_2$. The tensor product has roots $x_i+y_j$. Expanding their six pairwise products gives the displayed formula. This integral computation does not divide by two, so it remains valid when the <cohomology> has two-torsion. In particular, taking $F$ trivial gives $c_2(E\oplus E)=c_1(E)^2+2c_2(E)$.
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