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 (L^3) 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).