The Jacobi identity is the name for the following equation:
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0 for all X,Y,Z.
Lie algebras are the primary example of an algebra which satisfies the Jacobi identity. But note that an algebra can satisfy the Jacobi identity but yet not be anticommutative.
See also superjacobi identity.