  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 Lie algebra of maximal class over $E$ which are ideally
$r$-constrained for a positive integer $r$.
