Geodesic
Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices
Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 49-70

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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 $\mathrm{var}(sl_2(K))$.
Keywords: free Lie algebras, generic matrices, inner automorphisms, Baker–Campbell–Hausdorff formula. 
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     author = {Vesselin Drensky and \c{S}ehmus F{\i}nd{\i}k},
     title = {Inner automorphisms of {Lie} algebras related with generic $2\times 2$ matrices},
     journal = {Algebra and discrete mathematics},
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     number = {1},
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     url = {http://geodesic.mathdoc.fr/item/ADM_2012_14_1_a5/}
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Vesselin Drensky; Şehmus Fındık. Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices. Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 49-70. http://geodesic.mathdoc.fr/item/ADM_2012_14_1_a5/