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In this paper we study crystallographic sphere packings and Kleinian spherepackings, introduced first by Kontorovich and Nakamura in 2017 and then studiedfurther by Kapovich and Kontorovich in 2021. In particular, we solve theproblem of existence of crystallographic sphere packings in certain higherdimensions posed by Kontorovich and Nakamura. In addition, we present ageometric doubling procedure allowing to obtain sphere packings from someCoxeter polyhedra without isolated roots, and study ``properly integral''packings (that is, ones which are integral but not superintegral). Ourtechniques rely extensively on computations with Lorentzian quadratic forms,their orthogonal groups, and associated higher-dimensional hyperbolicpolyhedra.

Kleinian sphere packings, reflection groups, and arithmeticity