Kleinfeld function

Definition
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring) with multiplication $$*$$. Denote by $$a$$ the defining ingredient::associator of $$R$$, i.e., $$a(x,y,z) := ((x * y) * z) - (x * (y * z))$$. The Kleinfeld function of $$R$$ is a function $$f$$ from $$R^4$$ to $$R$$ defined as follows:

$$f(w,x,y,z) := a(w * x, y, z) - (x * a(w,y,z)) - (a(x,y,z) * w)$$

The Kleinfeld function is typically studied in the context of alternative rings, and it is used in one of the proofs of Artin's theorem on alternative rings.

Facts

 * Kleinfeld function is alternating in all pairs of variables for alternative ring