Jacobi Identity

Let $+,[-,-]:A\times A\to A$ be a pair of Binary Operations, and $0$ a 2-sided unit for addition. These two operations are said to satisfy the Jacobi Identity if:

• $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$

Note that each of the successive brackets has been rotated left by 1; in fact, these are precisely the Even Permutations of $(x,y,z)$.