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The quaternions are non-commutative. The deviation from commutativity isencapsulated in the commutator of unit quaternions. It is known that the k-thpower of the commutator is null-homotopic if and only if k is divisible by 12.The main purpose of this paper is to construct a concrete null-homotopy of the12-th power of the commutator. Subsequently, we construct free S^3-actions onS^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanationfor the order of the stable homotopy groups \pi_{n+3} (S^n). Intermediateresults of perhaps independent interest are a construction of the octonionsemphasizing the inclusion SU(3) \subset G_2, a detailed study of Duran'sgeodesic boundary map construction, and explicit formulas for thecharacteristic maps of the bundles G_2 \to S^6 and Spin(7) \to S^7.

On the commutator of unit quaternions and the numbers 12 and 24