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Let $\eta_2$ be the generator of $\pi_3(S^2)$, i.e. the Hopf map. $i_1:S^2 \hookrightarrow S^2\vee S^2$ and $i_2:S^2 \hookrightarrow S^2\vee S^2$ are the inclution maps respectively. then $[i_1\circ \eta_2,i_2]=?$ where $[,]$ is the whitehead product.

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Have you looked at the Hilton-Milnor theorem? It answers much more than your question. – Ryan Budney Mar 31 '12 at 8:29
By the Jocobi identity, we have: $$[i_1,[i_1,i_2]]+[i_1,[i_2,i_1]]+[i_2,[i_1,i_1]]=0$$ in which $[i_1,i_2]=[i_2,i_1]$ and $[i_1,i_1]=i_1\circ \eta_2$\\ i.e. $$2[i_1,[i_1,i_2]]+2[i_2,i_1\circ \eta_2]=0$$ Therefore, the difference between $[i_1,[i_1,i_2]]$ and $[i_2,i_1\circ \eta_2]$ is the 2 torsion part – jinch Apr 9 '12 at 8:10

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