  Let $K$ be a unital associative and commutative ring and let $K \langle X
\rangle$ be the free unital associative $K$-algebra on a non-empty set $X$ of
free generators. Define a left-normed commutator $[a_1, a_2, \dots , a_n]$
inductively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots , a_{n-1}, a_n] =
[[a_1, \dots , a_{n-1}], a_n]$ $(n \ge 3)$. For $n \ge 2$, let $T^{(n)}$ be the
two-sided ideal in $K \langle X \rangle$ generated by all commutators
$[a_1,a_2, \dots , a_n]$ $( a_i \in K \langle X \rangle )$.
  It can be easily seen that the ideal $T^{(2)}$ is generated (as a two-sided
ideal in $K \langle X \rangle$) by the commutators $[x_1, x_2]$ $(x_i \in X)$.
It is well-known that $T^{(3)}$ is generated by the polynomials $[x_1,x_2,x_3]$
and $[x_1,x_2][x_3,x_4] + [x_1,x_3][x_2,x_4]$ $(x_i \in X)$. A similar
generating set for $T^{(4)}$ contains 3 types of polynomials in $x_i \in X$ if
$\frac{1}{3} \in K$ and 5 types if $\frac{1}{3} \notin K$. In the present
article we exhibit a generating set for $T^{(5)}$ that contains 8 types of
polynomials in $x_i \in X$.
