  Recently Machado and Koshlukov have computed the Gelfand-Kirillov dimension
of the relatively free algebra $F_m=F_m(\text{var}(sl_2(K)))$ of rank $m$ in
the variety of algebras generated by the three-dimensional simple Lie algebra
$sl_2(K)$ over an infinite field $K$ of characteristic different from 2. They
have shown that $\text{GKdim}(F_m)=3(m-1)$. The algebra $F_m$ is isomorphic to
the Lie algebra generated by $m$ generic $2\times 2$ matrices. Now we give a
new proof for $\text{GKdim}(F_m)$ 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 $2\times 2$ matrices.
