I try to prove that higher homotopy groups of kan complexes are abelian using an Eckmann-Hilton argument. For the definitions I followed the book "Simplicial objects in algebraic Topology" by Peter May. He also has a proof for commutativity but I want to do it using an Eckmann-Hilton Argument.
Therefore I defined a second multiplication $\circledast$. For $[x], [y] \in \pi_n(K, \phi)$ the simplices $x, \hat{\;}, y, \underbrace{\phi, \dots, \phi}_{n-1-\text{times}}$ are compatible and we can find $w$ such that $d_i w = \phi$ for $2 \leq i \leq n + 1$ and $d_0 w = x, d_2 w = y$. For that we define $[x] \circledast [y] = [d_n w]$. The multiplication from the book which was similar but with $\underbrace{\phi, \dots, \phi}_{n-1-\text{times}},x, \hat{\;}, y$ will be denoted by $\otimes$.
For Eckman-Hilton we have to prove that for $[a],[b],[c],[d] \in \pi_n(K, \phi)$ it holds:
$([a] \otimes [b]) \circledast ([c] \otimes [d]) = ([a] \circledast [c]) \otimes ([b] \circledast [d])$
My current approach was to calculate both sides of the equation, but I did not manage to find a homotopy between the results to show that they are in the same equivalence class. Any hints on how to prove this is appreciated.