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...

In the present paper we generalize a result of a theorem of J. Jakubík concerning graph automorphisms of lattices to the case of multilattices of locally finite length.