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For based continuous maps $\alpha :S^p\to G$ and $\beta :S^q\to G$ into a topological group, take the pointwise commutatorIt is exactly the identity on $S^p\vee S^q\subseteq S^p\times S^q$, so it descends to the smash product $S^p\wedge S^q\cong S^{p+q}$. Its homotopy class is the Samelson product $⟨\alpha ,\beta ⟩\in {\pi }_{p+q}(G)$. This measures failure of the two maps to commute; it vanishes for an abelian group.^[ref]

For an ordinary loop space, reversal is an inverse up to homotopy, and a retraced loop need not be the constant loop. Literal descent therefore requires a strict topological group model or the corresponding coherent contractions, rather than treating loop reversal as a strict inverse. Equivalently, suspension-loop adjunction defines the operation through the sign-adjusted Whitehead product in the original space.

In positive degrees these operations give the graded antisymmetry and Jacobi identity of the loop-space homotopy Lie algebra. Under the Hurewicz homomorphism, the product becomes the graded commutator in the Pontryagin ring.

Tse Leung So studies homotopy-theoretic questions about Lie groups. His paper on commutator orders examines the Samelson product in low-rank groups and its odd-primary behavior.^[ref]

The group commutator is trivial when either input is the identity. It therefore descends from a Cartesian product to a smash product, turning noncommutativity into a based topological map. Its homotopy order measures information that the ordinary multiplication table alone does not express.^[ref]