First I will quote Adam's theorem (also known as the Hopf invariant theorem) from here:
"The Hopf invariant one theorem, sometimes also called Adams' theorem, is a deep theorem in homotopy theory which states that the only n-spheres which are H-spaces are $\mathbb S^0, \mathbb S^1, \mathbb S^3,$ and $\mathbb S^7$. The theorem was proved by Adams (1958, 1960)."
Now in the definition of H spaces it says (here):
An H-space, named after Heinz Hopf, and sometimes also called a Hopf space, is a topological space together with a continuous binary operation $\mu:X×X \rightarrow X,$ such that there exists a point $e$ in $X$ with the property that the two maps $x\mapsto \mu(x,e)$ and $x\mapsto \mu(e,x)$ are both homotopic to the identity map $id_X$ on $X$, through homotopies preserving the point $e$. The element e is called a homotopy identity.
Now my question is: What is the binary operation $\mu$ on those spheres? I assume it has something to to with Hurwitz Theorem/the induced cross product from the Real Numbers, the Complex Numbers, the quaternions and the octonions but I can't figure out $\mu $ itself. I tried searching for the answer but I couldn't find it.
I hope you can help me