Geodesic
On the theory of mainstay parallelohedra
Izvestiya. Mathematics , Tome 69 (2005) no. 6, pp. 1257-1277

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In 1998 the first author announced a theorem stating that every primitive $n$-dimensional parallelohedron can be represented, up to an affine transformation, as a weighted Minkowski sum of parallelohedra belonging to a certain finite set of $n'$-dimensional $(n'\leqslant n)$ mainstay parallelohedra situated in a special way. This paper contains a detailed proof of this theorem in a refined and definitive form. 
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S. S. Ryshkov; E. A. Bolshakova. On the theory of mainstay parallelohedra. Izvestiya. Mathematics , Tome 69 (2005) no. 6, pp. 1257-1277. http://geodesic.mathdoc.fr/item/IM2_2005_69_6_a9/