Thin Subalgebras of Lie Algebras of Maximal Class

For every field F which has a quadratic extension E we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension 2. We construct such Lie algebras as F-subalgebras of Lie algebras M of maximal class over E. We characterise the thin Lie F-subalgebras of M generated in degree 1. Moreover, we show that every thin Lie algebra L whose ring of graded endomorphisms of degree zero of L3 is a quadratic extension of F can be obtained in this way. We also characterise the 2-generator F-subalgebras of a Lie algebra of maximal class over E which are ideally r-constrained for a positive integer r.