It is well known that in a commutative ring $4xy = (x+y)^2 - (x-y)^2$.
Does such a formula exist for non-commutative rings? What I mean is:
Given any ring $R$ is there a $c\in \mathbb N$, affine functions $a_1,\dots, a_n : R^2 \to R$, $k_1,\dots, k_n \in \mathbb N$ and an affine function $b : R^n \to R$ s.t.
$$c(xy) = b(a_1(x,y)^{k_1}, \dots, a_n(x,y)^{k_n}) \text{ f.a. } x,y\in R?$$
My gut feeling is "no", but I don't have any idea how to prove something like this. Also, I don't know whether allowing affine instead of linear functions adds anything useful at all. What about special cases (e.g. $2\times 2$-matrices)?
Remark: of course, in non-zero characteristic the problem is trivial.