Inner Automorphisms of Lie Algebras Related with Generic 2 x 2 Matrices

Let F_m=F_m(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 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 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F_m/F_m^{c+1} in the subvariety of var(sl(2,K)) consisting of all nilpotent algebras of class at most c in var(sl(2,K)).