Explicitly, there are 112 roots with integer entries obtained from: formula_4by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from: formula_5by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).
Explicitly, there are 112 roots with integer entries obtained from: formula_16by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from: formula_17by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).