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.

This research proposes a straightforward and efficient method to optimize a sandwich monocoque car body with the developed equivalent shell element based on stiffness equivalence. The fact that stiffness rather than strength is dominant constraint for ordinary car body optimization is demonstrated. A simple but heavy flat chassis plate is utilized as upper bound, while an ideal monocoque is used as lower bound for an actual car body optimization. Convergence hours can be significantly reduced with the equivalent element and the initial-bounded method. A novel electric car body optimization is presented, fulfilling UltraLight Steel Auto Body (ULSAB) stiffness requirements and showing similar optimal weight results as a conventional approach with a much lower analysis time cost.