Koszul property
Some quadratic algebras have special property: Koszulity.
If we introduce nonhomogeneity in the Koszul quadratic algebra, it is enough to verify the Jacobi identity for three elementary letters. If it is satisfied, then the nonhomogeneous algebra will be automatically consistent, in the sense of the PBW property:
where is the corresponding homogeneous quadratic algebra.
In other words, the leading term is well-defined.