Endow $S^7$ with a structure of an $H$-space induced from multiplication in octonions $\mathbb{O}=\mathbb{R}^8$. It is not associative as octonion multiplication is not associative.


Is it associative up to homotopy, i.e. are maps $m(m(-,-),-):S^7\times S^7\times S^7\to S^7$ and $m(-,m(-,-)):S^7\times S^7\times S^7\to S^7 $ homotopic?



Is the $H$-space structure on $S^7$ associative up to homotopy?
It is not. See Theorem 1.4 of [this paper by I.M. James](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0090812-8/) (Trans. AMS 84 (1957), 545-558). 


In particular, there exists *no* homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$. 


