Initial simplicial cone in $n$-dimensional space have origin $(0, \dots, 0)$ and rays determined by vertices $(0, 0, \dots, 1), (0, 0, \dots, 1, 0) \dots (1, 0, \dots, 0)$. We dissect it by central hyperplane having normal vector with $n - 2$ zeros and $2$ ones (positive, negative or both) so that it is divided in two subsimplices. Then repeat dissection for each new subsimplex until cannot do it further. This can happen by two reasons:
1) no more crossing hyperplanes for each subsimplex - full dissection
2) one of subbodies is not simplex (more precisely simplicial cone)
Is there way to determine that on some step we chose "wrong" vector and cannot perform full dissection?
EDIT. It seems (by computer experiments) that full dissection is always attainable with these conditions. So I posted another question with original more complex problem. I don't know if this question still have some value or it should be closed.