Exponential lower estimate of the degree of nilpotency of Engel Lie algebras (Q1080510)

Let L be the free Lie algebra in two generators and \(L_ n=Z/I_ n\) where \(I_ n\) is the ideal of L generated by elements of the form [...[u,v], v]...,v] (n times). Therefore \(L_ n\) is the free 2-generated Engel Lie algebra of exponent n. According to Kostrikin's theorem \(L_ n\) is nilpotent. Let \(\gamma_ 2(n)\) be its index of nilpotency. The authors prove that \(\gamma_ 2(n)>2^{n/15}\) for \(n>N\) where N is a certain large enough natural number.