One of Jacobson’s most enduring contributions is the theory of (also called $p$-Lie algebras). He realized that in characteristic $p > 0$, the standard Lie bracket is insufficient; one must also include a $p$-th power map $x \mapsto x^[p]$, which behaves like the $p$-th power of a derivation. This structure is essential for linking Lie algebras to algebraic groups in positive characteristic.
. For a space to qualify as a Lie algebra, it must satisfy two fundamental properties: Cornell University Skew-symmetry , which implies Jacobi Identity for all elements Cornell University jacobson lie algebras pdf
In differential geometry, the TKJ construction explains the Lie algebra of the automorphism group of a bounded symmetric domain. Every Hermitian symmetric space corresponds to a Jordan triple system, whose associated Lie algebra is a Jacobson–Koecher–Tits algebra. The PDF by Loos (see below) is key here. One of Jacobson’s most enduring contributions is the