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.

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?