Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Recently Machado and Koshlukov have computed the Gelfand Kirillov dimension of the relatively free algebra F-m = F-m(var (sl(2)(K))) of rank m in the variety of algebras generated by the three-dimensional simple Lie algebra sl2 (K) over an infinite field K of characteristic different from 2. They have shown that GKdim(F-m) = 3(m-1). The algebra F-m is isomorphic to the Lie algebra generated by m generic 2 x 2 matrices. Now we give a new proof for GKdim(F-m) using classical results of Procesi and Razmyslov combined with the observation that the commutator ideal of F-m is a module of the center of the associative algebra generated by m generic traceless 2 x 2 matrices.89125135Computational and Combinatorial Methods in Algebra and Applications of the Bulgarian National Science Fund [I02/18]FAPESP [2014/08608-0]CNPq [304632/2015-5]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)