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Recently Machado and Koshlukov have computed the Gelfand-Kirillov dimensionof the relatively free algebra $F_m=F_m(\text{var}(sl_2(K)))$ of rank $m$ inthe variety of algebras generated by the three-dimensional simple Lie algebra$sl_2(K)$ over an infinite field $K$ of characteristic different from 2. Theyhave shown that $\text{GKdim}(F_m)=3(m-1)$. The algebra $F_m$ is isomorphic tothe Lie algebra generated by $m$ generic $2\times 2$ matrices. Now we give anew proof for $\text{GKdim}(F_m)$ using classical results of Procesi andRazmyslov combined with the observation that the commutator ideal of $F_m$ is amodule of the center of the associative algebra generated by $m$ generictraceless $2\times 2$ matrices.

GK-dimension of the Lie algebra of generic $2\times 2$ matrices