Integral elements of Okubo algebra and the E8-lattice

Abstract

In this work we study the interplay between the Coxeter-Dickson $E_{8}$-order, the para-octonions, and the real Okubo algebra. We prove that the Coxeter-Dickson order remains closed for the para-octonionic product, so that one recovers a genuine $\mathbb{Z}$-integral system with underlying lattice $E_{8}$. Intriguingly, the Okubo product behaves in a different and more arithmetic way: it forces $\mathbb{Q}(\sqrt{3})$-coefficients and does not preserve the same $\mathbb{Z}$-order. After a diagonal $2$-adic scaling we obtain a closed $\mathbb{Z}[\sqrt{3}]$-order, whose direct metric shadow is a $2$-primary conductor sublattice of $E_{8}$, not $E_{8}$ itself. The lattice $E_{8}$ is recovered only by $2$-adic saturation, equivalently by gluing, and this recovery is metric-arithmetic rather than multiplicative.