In his paper, Note on quasi-Lie rings, P. J. Hilton defines the (non-associative) Homotopy ring of a pointed space $X$ as$$\bigoplus_{n>1}\pi_n(X)$$ where the Whitehead product $\pi_m(X)\times\pi_n(X)\to\pi_{m+n-1}(X)$ is the product.

I suppose in a naïve fashion, one would hope for this to sprout some interesting structure in a similar way to cohomology rings. However it seems to be the case the non-associativity (stemming from the Whitehead product) kills this dream. As far as I can tell, no interesting ring theory can be done on this object, not to mention the fact that it is highly non-trivial for most $X$.

I failed to find anymore literature on this subject, partly due to terminology changing from one source to the next. I suppose the correct terminology would be to call this a quasi-Lie algebra, but this just doesn't have the same ring to it.

• Is there anything more to be said about this construction?
• Is it wrong to try to force a structure on homotopy groups in this way?
• Are there similar constructions that have the same feel as cohomology rings but for homotopy?