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The adjoint of the commutator on $\Omega J_{p^n-1}S^2$ is the Whitehead product $\Sigma (\Omega J_{p^n-1}S^2\wedge\Omega J_{p^n-1}S^2)\rightarrow J_{p^n-1}S^2$, which would vanish if $\Omega J_{p^n-1}S^2$ were homotopy commutative. This map is non-trivial since it is detected on the very bottom cell by a non-vanishing cup product. In fact on the bottom cell we see that it is the Hopf map $-2\eta:S^3\rightarrow S^2\hookrightarrow J_{p^n-1}S^2$ which generates $\pi_3S^2_{(p)}$.