 Whitehead square

The Whitehead square of a class $\alpha \in {\pi }_{m}Y$ is $[\alpha ,\alpha ]\in {\pi }_{2m-1}Y$. For the identity class of an even-dimensional sphere, the Hopf invariant is two with the standard product-cell orientation: folding the two sphere summands makes the middle cohomology class of the mapping cone pull back to $a+b$ on $S^m\times S^m$, and $(a+b{)}^2=2ab$. For odd $m$, graded symmetry makes the square two-torsion, hence rationally zero.