 Samelson product

For sphere maps into a loop space or a group-like homotopy-associative space, take their pointwise commutator. It is trivial on the wedge sum of the two sphere factors, so descends to their smash product and defines $⟨a,b⟩\in {\pi }_{p+q}(\Omega Y)$. This product obeys graded antisymmetry and the graded Jacobi identity. Suspension-loop adjunction identifies it with the sign-adjusted Whitehead product. Under the Hurewicz homomorphism it becomes the graded commutator of the Pontryagin product.