A sandwich in thin lie algebras

A thin Lie algebra is a Lie algebra, graded over the positive integers, with its first homogeneous component of dimension two and generating, and such that each non-zero ideal of lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of (that is, the next diamond past) occurs in degree. We prove that if 5$]]>, then for some non-zero element of. In characteristic different from two this means is a sandwich element of. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.