Algebraic Lie algebras with finite grading

The paper presents a variant of the proof of local finite-dimensionality of Lie PI-algebras with an algebraic adjoint representation over fields of characteristic zero without the use of extremal elements, a number of similar conclusions for such algebras over fields of characteristic $p> 7$, and generalizes the description of the locally finite radical of algebraic Mal'tsev locally PI-algebras to any base field of characteristic zero.