  Nonsingular derivations of modular Lie algebras which have finite
multiplicative order play a role in the coclass theory for pro-$p$ groups and
Lie algebras. A study of the set N_p of positive integers which occur as orders
of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of
positive characteristic p was initiated by Shalev and continued by the present
author. In this paper we continue this study in the case of characteristic two.
Among other results, we prove that any divisor n of 2^k-1 with
$n^4>(2^k-n)^{3}$ belongs to N_2. Our methods consist of elementary arguments
with polynomials over finite fields and a little character theory of finite
groups.
