A sandwich in thin Lie algebras

Abstract: A thin Lie algebras is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each nonzero ideal of $L$ lies between consecutive terms of its lower central series. All its homogeneous components have dimension one or two, and the two-dimensional components are called diamonds. We prove that if the next diamond past $L_1$ of an infinite-dimensional thin Lie algebra $L$ is $L_k$, with $k>5$, then $[Lyy]=0$ for some nonzero element $y$ of $L_1$.