We have spheroids $f: S^k\to X$ and $g: S^l \to X$. There is a property that $[f,g]=0$ in $\pi_{k+l-1}(X)$ iff map $f\vee g: S^k \vee S^l\to X$ extends to map $S^k\times S^l\to X$.
There was no explanation of why it is so, where I read this. Perhaps, it means that it is not too hard, but I still can't see why.
Thank you.
$S^k\times S^l$ is constructed by attaching a $k+l$-cell to $S^k\vee S^l$. The Whitehead product is defined using the attaching map $S^{k+l-1}\to S^k\times S^l$. If this map is trivial, then a nullhomotopy defines an extension to the $k+l$-cell, and conversely.