Geodesic
On the index of the boundary points of four-dimensional lattices that are admissible for cube
Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 176-185

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Let $\Gamma$ be a four-dimensional lattice of general position that is admissible for a cube. Assume that this lattice contains at least one point that belongs to the boundary of this cube. We prove that the index of the set of such points may be equal only to 0, 1, and 2. 
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     author = {Yu. A. Smirnov},
     title = {On the index of the boundary points of four-dimensional lattices that are admissible for cube},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {176--185},
     publisher = {mathdoc},
     volume = {322},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a11/}
}
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Yu. A. Smirnov. On the index of the boundary points of four-dimensional lattices that are admissible for cube. Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 176-185. http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a11/