How do one prove (or disprove) that $\Omega S^{2}$ and $\Omega(S^{2} \vee S^{2})$ are non commutative Hopf spaces?

I thought this is a question for math.stackexchange, but not many people even viewed that question. Seems like it should be easy but right now I have no idea how to do it. Any hint?

Definition: If $X$ is commutative Hopf space if, for the multiplication $ \mu $ there is a homotopy between $ H: \mu \simeq \mu \circ T $, where $T : X \times X \to X \times X$ is the switch map.

Clarification: When I say multiplication, I mean multiplication induced by the loop structure. On $\Omega S^{2}$ there is another multiplication which is induced by multiplication on $S^{1}$ as $\Omega S^{2} = \Omega \Sigma S^{1} $ ( This is of course commutative).