Once you have a PDF (say, Chapter IX of Jacobson's book), you will face dense notation. Here is a reading strategy.
To understand Jacobson Lie algebras, one must first appreciate the work of Nathan Jacobson (1910–1999), one of the foremost algebraists of the 20th century. Jacobson is best known for his groundbreaking work on ring theory, particularly the Jacobson radical—a concept that identifies the "bad" (nilpotent or quasi-regular) elements of a ring. jacobson lie algebras pdf
The transition from rings to Lie algebras occurs naturally: many properties of associative rings can be mirrored in Lie algebras via the commutator bracket ([x, y] = xy - yx). A Lie algebra is called Jacobson (or more precisely, a Jacobson Lie algebra) if it satisfies certain nilpotency or radical conditions analogous to the Jacobson radical in associative rings. However, terminology can vary. In some contexts, a "Jacobson Lie algebra" refers to a Lie algebra whose adjoint representation is Jacobson (i.e., every element is ad-nilpotent or the algebra is locally nilpotent). In other sources, it aligns with the study of Lie algebras with a nilpotent Jacobson radical of their universal enveloping algebra. Once you have a PDF (say, Chapter IX
Let ( L ) be a Lie algebra consisting of nilpotent linear transformations on a finite-dimensional vector space over a field of characteristic zero. Then ( L ) is nilpotent. Let ( L ) be a Lie algebra
This theorem, proved by Jacobson in 1955, is a cornerstone. It is often included in advanced Lie algebra lecture notes (available as PDFs from university repositories).