The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

Let Z〈X〉Z〈X〉 be the free unital associative ring freely generated by an infinite countable set X={x1,x2,…}X={x1,x2,…}. Define a left-normed commutator [a1,a2,…,an][a1,a2,…,an] inductively by [a,b]=ab−ba[a,b]=ab−ba, [a1,a2,…,an]=[[a1,…,an−1],an][a1,a2,…,an]=[[a1,…,an−1],an] (n≥3n≥3). For n≥2n≥2, let T(n)T(n) be the two-sided ideal in Z〈X〉Z〈X〉 generated by all commutators [a1,a2,…,an][a1,a2,…,an] (ai∈Z〈X〉ai∈Z〈X〉). Let T(3,2)T(3,2) be the two-sided ideal of the ring Z〈X〉Z〈X〉 generated by all elements [a1,a2,a3,a4][a1,a2,a3,a4] and [a1,a2][a3,a4,a5][a1,a2][a3,a4,a5] (ai∈Z〈X〉ai∈Z〈X〉).It has been recently proved in [22] that the additive group of Z〈X〉/T(4)Z〈X〉/T(4) is a direct sum A⊕BA⊕B where A   is a free abelian group isomorphic to the additive group of Z〈X〉/T(3,2)Z〈X〉/T(3,2) and B=T(3,2)/T(4)B=T(3,2)/T(4) is an elementary abelian 3-group. A basis of the free abelian summand A was described explicitly in [22]. The aim of the present article is to find a basis of the elementary abelian 3-group B.