What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$?  The group structure induced by the group structure on the codomain.  This question is a followup to Eric's answer to [another question][1].


Added later: The commutator map $(g,h) \mapsto ghg^{-1} h^{-1}$ descends to a map from $S^6$ to $S^3$.  Which element of $\pi_6(S^3) = \mathbf{Z}/12$ is this?

  [1]: https://mathoverflow.net/questions/160285/can-the-group-of-homotopy-classes-of-maps-into-s3-be-noncommutative