We provide an explicit construction for a class of commutative, non-associative algebras each the simple Chevalley groups simply laced type. Moreover, we equip these with associating bilinear form, which turns them into Frobenius algebras. This includes 3876-dimensional algebra on group type E8 acts by automorphisms. also prove that admit structure (axial) decomposition

We prove that if $$G(R)=G_\pi (\Phi ,R)$$ $$(E(R)=E_{\pi }(\Phi , R))$$ is an (elementary) Chevalley group of rank $$> 1$$ R a local ring (with $$\frac{1}{2}$$ for the root systems $${{\textbf{A}}}_2, {{\textbf{B}}}_l, {{\textbf{C}}}_l, {{\textbf{F}}}_4, {{\textbf{G}}}_2$$ and with $$\frac{1}{3}$$ $${{\textbf{G}}}_{2})$$ then G(R) (or (E(R)) regularly bi-interpretable R. As consequence this the...