Centre-by-metabelian group algebras

In nonabelian groups, as well as in noncommutative associative algebras, one may measure the degree of noncommutativity with the help of commutators. Based on these, one defines in „both worlds" (the category of groups, and the category of associative algebras) analogous concepts, such as (Lie) solvability, (Lie) nilpotence, … - here the question arises whether one also obtains parallel properties in both categories. In this context we take as the object of study the group algebra of a (finite or infinite) group over some field. A „parallel property" would then be e.g. commutativity: A group is abelian, if and only if the associated group algebra is commutative. For other, more complex concepts such as the ones mentioned above, such a total correlation cannot be expected, although it should be clear that the commutator properties of the group algebra are derived from the commutator properties of the group we started with.