Theorem on $L$-partitions of point lattices

It is shown that if a simplex $S$ is a basic $L$-simplex for a point lattice in $E^n$ ($n\le5$), then the lattice's $L$-simplexes that are contiguous to $S$ by $(n-1)$-faces can have as vertices lattice points belonging to a specified set of points $P(S)$, and a complete description of this set is given. Based on the fact that the set $P(S)$ is known, a new method of deriving the types of point lattices, different from the known methods (G. F. Voronoi's algorithm and B. N. Delaunay's method of layers), is obtained. The types of primitive lattices in $E^3$ and $E^4$ are derived by this method.