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Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice
We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E
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