University of IsfahanInternational Journal of Group Theory2251-76507220180601On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras45502148110.22108/ijgt.2017.21481ENNilMansuroğluAhi Evran UniversityJournal Article20161121Let $L$ be a free Lie algebra of rank $r\geq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.<br /><br />https://ijgt.ui.ac.ir/article_21481_0b392ad1ffab7cd79272442ecc91712c.pdf