The Okubo algebra's integral shells form a two-adic hierarchy of polytopes that decompose into W(B8) orbits on a rescaled cubic lattice, recovering the E8 Gosset polytope by maximal-isotropic gluing along (Z/2)^4.
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Para-octonionic product closes the Coxeter-Dickson E8-order for genuine Z-integral E8, but Okubo forces Q(sqrt(3)) coefficients; after 2-adic scaling a Z[sqrt(3)]-order shadows a 2-primary conductor sublattice of E8, recovered only by saturation or gluing as metric-arithmetic rather than purely by a
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Integral elements of Okubo algebra and the E8-lattice
Para-octonionic product closes the Coxeter-Dickson E8-order for genuine Z-integral E8, but Okubo forces Q(sqrt(3)) coefficients; after 2-adic scaling a Z[sqrt(3)]-order shadows a 2-primary conductor sublattice of E8, recovered only by saturation or gluing as metric-arithmetic rather than purely by a