Simplexes of $L$-subdivisions of Euclidean spaces

It is shown that necessary and sufficient conditions for a basic simplex of a point lattice in $E^n$ space to be an $L$-simplex are equivalent to conditions imposed on the coefficients $a_{ij}$ of the form $\sum_{i,j=1}^na_{ij}x_ix_j-\sum_{i=1}^na_{ii}x_i$, namely, that it should assume positive values for all possible integer values of the variables $x_1,\dots,x_n$ (excluding the obvious $n+1$ cases when the form is equal to 0). These conditions are obtained for $n\leqslant5$.