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Define a left-normed commutator $[a_1, a_2, \\dots, a_n]$ recursively 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 $F \\langle X \\rangle$ generated by all commutators $[a_1, a_2, \\dots, a_n]$ ($a_i \\in F \\langle X \\rangle)$.\n  Let $F$ be a field of characteristic $0$. 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