Constituents of graded Lie algebras of maximal class and chains of thin Lie algebras

A thin Lie algebra is a L, graded over the positive integers, with its first homogeneous component L1 of dimension two and generating such that each nonzero ideal L lies between consecutive terms lower central series. All components have one or two, two-dimensional are called diamonds. If only diamond, then maximal class. We present simpler proofs some fundamental facts on algebras class, algebras, based uniform method, emphasis polynomial interpretation. Among else, we determine possible values for most parameter which less than their largest metabelian quotient.