Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices

Let $F_m=F_m(\mathrm{var}(sl_2(K)))$ be the relatively free algebra of rank $m$ in the variety of Lie algebras generated by the algebra $sl_2(K)$ over a field $K$ of characteristic $0$. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion $\widehat{F_m}$ of $F_m$ with respect to the formal power series topology. Our results are more precise for $m=2$ when $F_2$ is isomorphic to the Lie algebra $L$ generated by two generic traceless $2\times 2$ matrices. We give a complete description of the group of inner automorphisms of $\widehat L$. As a consequence we obtain similar results for the automorphisms of the relatively free algebra $F_m/F_m^{c+1}=F_m(\mathrm{var}(sl_2(K))\cap {\mathfrak N}_c)$ in the subvariety of $\mathrm{var}(sl_2(K))$ consisting of all nilpotent algebras of class at most $c$ in $\mathrm{var}(sl_2(K))$.