: Jacobson provided a rigorous treatment of this theorem, which ensures that contains a copy of the original Lie algebra and provides a basis for studying its representations .
In characteristic 0, Engel’s theorem states that if every element of a Lie algebra is ad-nilpotent, the algebra is nilpotent. Jacobson extended this to characteristic $p$ with a crucial twist: If $L$ is a Lie algebra over a field of characteristic $p > 0$ and $x^p$ (the $p$-th power in the universal enveloping algebra) acts nilpotently for all $x$, then $L$ is nilpotent. This is often called . jacobson lie algebras pdf
These theorems are the bedrock of the field. Jacobson’s proofs are celebrated for their clarity: : Jacobson provided a rigorous treatment of this