Thin subalgebras of Lie algebras of maximal class

For every field $$\mathbb{F}$$ which has a quadratic extension $$\mathbb{E}$$ 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 $$\mathbb{F}$$ -subalgebras of Lie algebras M of maximal class over $$\mathbb{E}$$ . We characterise the thin Lie $$\mathbb{F}$$ -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 $$\mathbb{F}$$ can be obtained in this way. We also characterise the 2-generator $$\mathbb{F}$$ -subalgebras of a Lie algebra of maximal class over $$\mathbb{E}$$ which are ideally r -constrained for a positive integer r .