  Let $\mathbb Z \langle X \rangle$ be the free unital associative ring freely
generated by an infinite countable set $X = \{ x_1,x_2, \dots \}$. Define a
left-normed commutator $[x_1,x_2, \dots, x_n]$ by $[a,b] = ab - ba$, $[a,b,c] =
[[a,b],c]$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $\mathbb Z
\langle X \rangle$ generated by all commutators $[a_1,a_2, \dots, a_n]$ $( a_i
\in \mathbb Z \langle X \rangle )$. Let $T^{(3,2)}$ be the two-sided ideal of
the ring $\mathbb Z \langle X \rangle$ generated by all elements $[a_1, a_2,
a_3, a_4]$ and $[a_1, a_2] [a_3, a_4, a_5]$ $(a_i \in \mathbb Z \langle X
\rangle)$.
  It has been recently proved in arXiv:1204.2674 that the additive group of
$\mathbb Z \langle X \rangle / T^{(4)}$ is a direct sum $ A \oplus B$ where $A$
is a free abelian group isomorphic to the additive group of $\mathbb Z \langle
X \rangle / T^{(3,2)}$ and $B = T^{(3,2)} /T^{(4)}$ is an elementary abelian
$3$-group. A basis of the free abelian summand $A$ was described explicitly in
arXiv:1204.2674. The aim of the present article is to find a basis of the
elementary abelian $3$-group $B$.
