The structure of Lie algebras with a derivation satisfying a polynomial identity

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a periodic derivation of order $n$. We find an optimal bound on the nilpotency class in characteristic $p$ if $p$ does not divide a certain invariant $\rho_n$. We give a new description of the set $\mathcal{N}_p$ of positive integers $n$, introduced by Shalev, which arise as the order of a periodic derivation of a finite-dimensional non-nilpotent Lie algebra in characteristic $p>0$. Finally we generalize the results to Lie rings over $\Bbb Z$.