A test for commutators

In the course of a study of commutator subgroups I. D. Macdonald [1] presented the free nilpotent group G 4 of class 2 on 4 generators as an example of a nilpotent group whose commutator subgroup has elements that are not commutators. To demonstrate this he proceeded as follows: let G 4 = 〈 a 1 , a 2 , a 3 , a 4 〉 and put c ij = [a i , a j ] for 1 ≦ i j ≦ 4. Then the relations in G 4 are [ c ij , a k ] = 1 for 1 ≦ i j ≦ 4 and 1 ≦ k ≦ 4, and their consequences. Macdonald observed that an arbitrary commutator may be written as which simplifies to where δ ij = α i β j - α j β i The indices δ ij satisfy the relation It follows that the element c 13 c 24 in G′4 (for which δ 12 = δ 14 = δ 23 = δ 34 = 0 and δ 13 = δ 24 = 1) is not a commutator.