The nilpotency class of the sandwich subalgebra of simple finite-dimensional Lie algebras

In this paper it is shown that the nilpotency class of sandwich subalgebras in Lie algebras of Cartan type and in Melikyan algebras over a field of characteristic $p\geqslant 5$ is equal to $2m-1$, where $m$ is the sum of the heights of the variables. The only exception is the Zassenhaus algebra (and also the Hamiltonian and contact algebras for $p=5$).