A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand, a quasi-Lie algebra is defined by replacing the axiom $[x,x]=0$ with the antisymmetry axiom $[x,y]+[y,x]=0$ for all $x$ and $y$. Jerry Levine used the free quasi-Lie algebra to study the group of homology cylinders. See this paper and this paper. My question is whether there are other contexts in which quasi-Lie algebras appear. Lie algebras arise naturally in all sorts of different contexts, but I don't know of many places quasi-Lie algebras appear. For example, if one looks at the associated graded object for the lower central series of a group, one naturally obtains a Lie algebra and not a quasi-Lie algebra.

One possible answer is this MO post, where it is noted that in operad language, the quasi-Lie axioms are necessary, and this is quite germane to why Levine needed to use quasi-Lie algebras, since he was essentially building a group out of the Lie operad. But I'd like to know of any other examples where quasi-Lie algebras occur that people might know.