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A thin Lie algebra 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 non-zero ideal of $L$
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 $L$
(that is, the next diamond past $L_1$
) occurs in degree $k$
. We prove that if $k>5$
, then $[Lyy]=0$
for some non-zero element $y$
of $L_1$
. In characteristic different from two this means $y$
is a sandwich element of $L$
. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.
