Exponentials in Lie algebras of characteristic $p$

The relationship between the structure of a simple Lie algebra of finite characteristic and the structure of the group of its automorphisms is investigated. The results obtained are used to classify simple Lie algebras of characteristic $p>5$ for which the largest reduced subgroup in the scheme of automorphisms is a maximal subscheme. An analogous classification theorem is proved for “simple” group schemes, i.e. schemes every normal divisor of which lying in the reduced subscheme is the kernel of some purely nonseparable isogeny. For characteristics 2 and 3, families of counterexamples are constructed to all results obtained for $p>5$.