The definition and terminology of a restricted Lie algebra was developed by N. Jacobson as a method to mimic the properties of Lie algebras of characteristic 0. The usual methods of using minimal polynomials to establish the Jordan normal form of a transform fail in positive characteristic as certain polynomials become inseparable or reducible. Thus one cannot simply establish the typical nilpotent+semisimple decomposition of elements.

However, given a linear Lie algebra (subalgebra of $\mathfrak{gl}_k V$ ) over a field $k$ of positive characteristic $p$ then the map $x\mapsto x^p$ on the matrices captures many of the properties of the field. For example, a diagonal matrix $D$ with entries in $\mathbb{Z}_p$ satisfies $D^p=D$ simply because the power map $p$ is a field automorphism. Thus in various ways the power map captures the requirements of semisimple and toral elements of Lie algebras. By modifying the definitions of semisimple, toral, and nilpotent elements to use the power maps of the field, Jacobson and others were able to reproduce much of the classical theory of Lie algebras for restricted Lie algebras.

The definition given above reflects the abstract requirements for a restricted Lie algebra. However an important observation is that given a linear Lie algebra $L$ over a field of characteristic $p$ , then the usual associative power map serves as a power map for establishing a restricted Lie algebra. The only added requirement is that $L^p\subseteq L$ .

Jacobson, Nathan Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.

Strade, Helmut and Farnsteiner, Rolf Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker Inc., New York, 1988.