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.