In his article on Simplicial Homotopy Theory (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The formula for the Samelson product $\langle x,y\rangle$ of elements $x\in \pi_p(G)$ and $y\in \pi_q(G)$ is in terms of an ordered product of commutators of pairs of certain degenerate elements. The order of multiplication is determined by an antilex total order on the set of $(p,q)$-shuffles. I tried, and think I succeeded in filling in the details of this sketch from Curtis. I understand that the result was due to Dan Kan but that he never published it.

My questions are

(i) is there a published proof of this formula and if so where? (As I said I have a proof that it seems to work, but do not yet have one that it actually deserves to be called the Samelson product and that the related Whitehead product corresponds to the classical form, e.g. involving a 'universal example'.)

and

(ii) I have so far failed to prove that this formula gives a bilinear pairing on homotopy groups. (It is simple to prove a related formula which incorporates an action as in the Witt-Hall identities for commutators but I do not see why the action involved should be trivial as would seem to be required for the bilinearity result to hold.)

Can anyone suggest a reference to a classical proof of bilinearity of the (topological) Whitehead product that is fairly categorical and hence adaptable to my simplicial setting. Most sources seem to leave it as an exercise and Whitehead's original approach although clear is very topological in nature.

(Edit: I note that Adams in his Student's Guide to Algebraic Topology, stated: The Whitehead product is bilinear and anticommutative; this can be proved by diagram chasing with the universal example'. Unfortunately I have yet to find the diagram through which I have to chase, although I have tried several ones that initially seemed to give some hope of being good for the task in hand.)