Proceedings of the American Mathematical Society
In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of p-power automorphisms of p-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.
Mathematics and Statistics
Feldvoss, J., Siciliano, S., & Weigel, T. (2013). Outer restricted derivations of nilpotent restricted Lie algebras. Proceedings of the American Mathematical Society, 141(1), 171-179.