Are there any natural non-antisymmetric binary operations that satisfy the Jacobi identity?

Question

A binary operation $\times$ on a commutative monoid is defined to satisfy the Jacobi identity if, for all $a$, $b$, and $c$ in the monoid, $$ a \times (b \times c) + c \times (a \times b) + b \times (c \times a) = 0.$$

The Wikipedia article gives several examples of common binary operations that obey the Jacobi identity: the cross product of vectors in $\mathbb{R}^3$, the Lie bracket, the Poisson bracket in classical mechanics, and the operator commutator and Moyal bracket in quantum mechanics. For all of these examples, the binary operation is antisymmetric (or at least alternating, technically). Are there any natural examples of binary operations that satisfy the Jacobi identity but aren't antisymmetric(/alternating)? By "natural," I mean "not cooked up specifically to be an example of a non-antisymmetric operator that satisfies the Jacobi identity" :-).

Answer 1

In the skew-symmetric context, the bracket $[x,[y,z]]$ can freely be rewritten as $-[x,[z,y]]$, $-[[y,z],x]$, $[[z,y],x]$... which gives many ways of writing the Jacobi identity. If skew-symmetry is dropped, this yields many non-equivalent ways to rewrite (or rather, to lift) the Jacobi identity. One way of lifting is of importance:

$$[[x,y],z]=[[x,z],y]+[x,[y,z]]$$

It means that right multiplications are derivations. A (non-associative) algebra satisfying this axiom is called (right) Leibniz algebra.

Answer 2

Jean-Louis Loday introduced Leibniz algebras (which are Lie algebras, without skew-symmetry, see Yves' answer), because it appears "naturally" in algebraic K-theory. Roughly speaking the Lie algebra homology is related to the appearance of cyclic homology.

Reference: Jean-Louis Loday, Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices, Comm. Math. Helv. 59, n. 1, 565-591 (1984).

Lie algebra homology involves the Chevalley-Eilenberg chain complex, which in turns involves the exterior powers of the Lie algebra. Loday found that there is a noncommutative generalization where one has the tensor and not the exterior powers of the Lie algebra in the complex; this new complex defines the Leibniz homology of Lie algebras.