The additive group of a Lie nilpotent associative ring

Let Z〈X〉 be the free unital associative ring freely generated by an infinite countable set X={x1,x2,…}. Define a left-normed commutator [x1,x2,…,xn] by [a,b]=ab−ba, [a,b,c]=[[a,b],c]. For n⩾2, let T(n) be the two-sided ideal in Z〈X〉 generated by all commutators [a1,a2,…,an] (ai∈Z〈X〉). It can be easily seen that the additive group of the quotient ring Z〈X〉/T(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z〈X〉/T(3) is also free abelian. In the present note we show that this is not the case for Z〈X〉/T(4). More precisely, let T(3,2) be the ideal in Z〈X〉 generated by T(4) together with all elements [a1,a2,a3][a4,a5] (ai∈Z〈X〉). We prove that T(3,2)/T(4) is a non-trivial elementary abelian 3-group and the additive group of Z〈X〉/T(3,2) is free abelian.