# Article

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Keywords:
integral octonion; 24-cell; Gosset polytope
Summary:
We study the integral quaternions and the integral octonions along the combinatorics of the \$24\$-cell, a uniform polytope with the symmetry \$D_{4}\$, and the Gosset polytope \$4_{21}\$ with the symmetry \$E_{8}\$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope \$4_{21}\$ or a \$24\$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the \$E_{8}\$ or \$D_{4}\$ actions on the \$4_{21}\$ or the \$24\$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope \$2_{31}\$ with the symmetry \$E_{7}\$, and it is a dual polytope to a Gosset polytope \$3_{21}\$ with the symmetry \$E_{7}\$ which is the set of the unit integral octonions with \$\operatorname {Re}=1/2\$.
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