Restricted Lie algebras all whose elements are semisimple (Q2430821)

The authors consider restricted Lie algebras that consist of elements semisimple with respect to a \(p\)-mapping. It is known that a finite dimensional restricted Lie algebra with no nil \(p\)-elements over an algebraically closed field is abelian [\textit{B. Chwe}, Proc. Am. Math. Soc. 16, 547 (1965; Zbl 0135.07301)]. \textit{R. Farnsteiner} showed that in case of a not algebraically closed field the situation is more complicated [J. Algebra 83, 510--519 (1983; Zbl 0521.17003)]. Also \textit{R. Farnsteiner} studied the case of a semilinear \(p\)-mapping [Proc. Am. Math. Soc. 91, 41--45 (1984; Zbl 0511.17004)]. The field assumed to be perfect, the authors use these results of Farnsteiner and discuss different conditions on restricted Lie algebras with semisimple elements that imply commutativity. From the abstract: The authors ``show that a central-simple restricted Lie algebra all whose elements are semisimple over a field of characteristic \(p>7\) is a form of a classical Lie algebra''.