I agree that Archimedes was amazing.

@Ryan: is the fact that geometry is less useful in high dimensions an empirical observation, or is there more mathematical content to this?

@Mariano: Thanks for the dictionary translating Hilton's terms into modern ones. I have to admit I didn't read it carefully, mainly because he finds both 2 and 3-torsion in the free quasi Lie ring, whereas there is only 2-torsion in Levine's free quasi-Lie algebra. So I concluded they are not the same definition. I think you are right that taking $M=0$ in Hilton's definition does give you a quasi Lie algebra in Levine's sense, although I'm still unsure where that 3-torsion is coming from.

This raises an interesting question of what happens over $\mathbb Z_2$.

Tom gave a perfectly fine answer, so I'm posting it as a community wiki and accepting it.

@Tom: you probably didn't put this in an answer since it is awfully simple to be getting reputation points for, but it is a genuine answer...

Unfortuntately, the term "quasi-Lie ring" used in this reference does not mean the same thing as Levine's term. However, the odd homotopy groups with Whitehead bracket does form a quasi-Lie algebra under Levine's definition.

@Daniel: I even added a bounty.

@Daniel: I approve.

Hey, that guy looks awfully familiar.

@Bill: That's a clever way to do it!