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 .