On the dimension of the product [L2,L2,L1] in free Lie algebras

Let L be a free Lie algebra of rank r ≥ 2 over a field F and let Ln denote the degree n homogeneous component of L. By using the dimensions of the corresponding homogeneous and ne homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field F, we determine the dimension of [L2;L2;L1]. Moreover, by this method, we show that the dimension of [L2;L2;L1] over a eld of characteristic 2 is different from the dimension over a eld of characteristic other than 2.