This may be too specific for what you're going for, but this question immediately brought to mind to triality on $D_4$. Here's triality in a nutshell:
View $\mathbb{R}^8 = \mathbb{O}$ as the inner product space collection of all Cayley numbers (where $\{1,i,j,k,l,li,lj,lk\}$ forms an orthonormal basis.). The collection of all orientation preserving isometries of $\mathbb{O}$ is simply $SO(8)$ (whose Lie algebra is $D_4$).
For $A$, $B$, and $C$ in $SO(8)$ and $x$ and $y$ in $\mathbb{O}$, consider the equation $$A(x)B(y) = C(xy)$$
Triality is the following claim: Given $A$, there exists a $B$ and $C$ making this equation hold for all Cayley numbers. The choice of the pair (B,C) is ALMOST unique - the only ambiguity is in replacing $(B,C)$ with $(-B,-C)$. Likewise, given $B$ or $C$, the other two matrices exist and are unique up to simultaneously changing the sign of both.
This gives rise to a natural relation $R\subseteq SO(8)\times SO(8)\times SO(8)$ with $(A,B,C)\in R$ iff $A(x)B(y) = C(xy)$ for all Cayley numbers $x$ and $y$. The ambiguity in sign shows that this is not a 2-ary function.
