A ring $A$ that satisfies the conditions
(the Jacobi identity), where $a,b,c$ are any elements of $A$. The first of these conditions implies that $A$ is anti-commutative:
The Lie rings form a variety of rings, in general non-associative. It contains, however, all rings with zero multiplication.
See also Non-associative rings and algebras.
Lie ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_ring&oldid=32508