Barns-Wall-Lattice and reflection

G = matrix.identity(16)
basis = [[2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,2,-2,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,0,0,0,0,0,0,2,2,0,0,0,0,0,0],[2,2,0,0,0,0,0,0,0,0,0,0,2,2,0,0],[2,0,2,0,0,0,0,0,2,0,2,0,0,0,0,0],[2,0,2,0,0,0,0,0,0,0,0,0,2,0,2,0],[2,0,0,0,0,0,0,-2,0,2,0,0,0,0,2,0],[2,0,0,0,0,0,0,-2,0,0,2,0,0,-2,0,0],[2,0,0,0,-2,0,0,0,2,0,0,0,-2,0,0,0],[2,0,0,0,-2,0,0,0,-2,0,0,0,2,0,0,0],[2,0,0,-2,0,0,0,0,2,0,0,-2,0,0,0,0],[2,0,0,-2,0,0,0,0,0,0,0,0,2,0,0,-2],[2,0,0,0,0,0,0,-2,0,0,-2,0,0,2,0,0],[2,0,2,0,0,0,0,0,0,0,0,0,-2,0,-2,0],[2,0,2,0,-2,0,-2,0,0,0,0,0,0,0,0,0],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]]
L = IntegralLattice(G, basis)
Aut = L.orthogonal_group()

#The basis
[[2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0],
 [2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0],
 [2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0],
 [2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0],
 [2, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 2, 0],
 [2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0],
 [2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0],
 [2, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0],
 [2, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0],
 [2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2],
 [2, 0, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, 2, 0, 0],
 [2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0],
 [2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]

#鏡映の定義
def refl(a):
    unit = []
    for i in range(len(a[0])):
        unit.append(matrix(matrix.identity(len(a[0]))[i]))
    unit2 = []
    for j in range(len(unit)):
        unit2.append((unit[j] - 2*a*(unit[j]*a.transpose())[0][0] /(a*a.transpose())[0][0]))
    return matrix([unit2[i][0] for i in range(len(a[0]))])

#reflections
[2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
---------ref---------->
[ 1/2 -1/2 -1/2 -1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2  1/2 -1/2 -1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2 -1/2  1/2 -1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2 -1/2 -1/2  1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
---------ref---------->
[ 1/2 -1/2 -1/2  1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2  1/2 -1/2  1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2 -1/2  1/2  1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[ 1/2  1/2  1/2  1/2    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0]
---------ref---------->
[ 1/2 -1/2    0    0    0    0    0    0 -1/2 -1/2    0    0    0    0    0    0]
[-1/2  1/2    0    0    0    0    0    0 -1/2 -1/2    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[-1/2 -1/2    0    0    0    0    0    0  1/2 -1/2    0    0    0    0    0    0]
[-1/2 -1/2    0    0    0    0    0    0 -1/2  1/2    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0]
---------ref---------->
[ 1/2 -1/2    0    0    0    0    0    0    0    0    0    0 -1/2 -1/2    0    0]
[-1/2  1/2    0    0    0    0    0    0    0    0    0    0 -1/2 -1/2    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[-1/2 -1/2    0    0    0    0    0    0    0    0    0    0  1/2 -1/2    0    0]
[-1/2 -1/2    0    0    0    0    0    0    0    0    0    0 -1/2  1/2    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0]
---------ref---------->
[ 1/2    0 -1/2    0    0    0    0    0 -1/2    0 -1/2    0    0    0    0    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2    0  1/2    0    0    0    0    0 -1/2    0 -1/2    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[-1/2    0 -1/2    0    0    0    0    0  1/2    0 -1/2    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[-1/2    0 -1/2    0    0    0    0    0 -1/2    0  1/2    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0]
---------ref---------->
[ 1/2    0 -1/2    0    0    0    0    0    0    0    0    0 -1/2    0 -1/2    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[-1/2    0  1/2    0    0    0    0    0    0    0    0    0 -1/2    0 -1/2    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[-1/2    0 -1/2    0    0    0    0    0    0    0    0    0  1/2    0 -1/2    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[-1/2    0 -1/2    0    0    0    0    0    0    0    0    0 -1/2    0  1/2    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 2, 0]
---------ref---------->
[ 1/2    0    0    0    0    0    0  1/2    0 -1/2    0    0    0    0 -1/2    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[ 1/2    0    0    0    0    0    0  1/2    0  1/2    0    0    0    0  1/2    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[-1/2    0    0    0    0    0    0  1/2    0  1/2    0    0    0    0 -1/2    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[-1/2    0    0    0    0    0    0  1/2    0 -1/2    0    0    0    0  1/2    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0]
---------ref---------->
[ 1/2    0    0    0    0    0    0  1/2    0    0 -1/2    0    0  1/2    0    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[ 1/2    0    0    0    0    0    0  1/2    0    0  1/2    0    0 -1/2    0    0]
[   0    0    0    0    0    0    0    0    1    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[-1/2    0    0    0    0    0    0  1/2    0    0  1/2    0    0  1/2    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    1    0    0    0]
[ 1/2    0    0    0    0    0    0 -1/2    0    0  1/2    0    0  1/2    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0]
---------ref---------->
[ 1/2    0    0    0  1/2    0    0    0 -1/2    0    0    0  1/2    0    0    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[ 1/2    0    0    0  1/2    0    0    0  1/2    0    0    0 -1/2    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[-1/2    0    0    0  1/2    0    0    0  1/2    0    0    0  1/2    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[ 1/2    0    0    0 -1/2    0    0    0  1/2    0    0    0  1/2    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0]
---------ref---------->
[ 1/2    0    0    0  1/2    0    0    0  1/2    0    0    0 -1/2    0    0    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    1    0    0    0    0    0    0    0    0    0    0    0    0]
[ 1/2    0    0    0  1/2    0    0    0 -1/2    0    0    0  1/2    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[ 1/2    0    0    0 -1/2    0    0    0  1/2    0    0    0  1/2    0    0    0]
[   0    0    0    0    0    0    0    0    0    1    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    1    0    0    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    1    0    0    0    0]
[-1/2    0    0    0  1/2    0    0    0  1/2    0    0    0  1/2    0    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    1    0    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    1    0]
[   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1]
************************
[2, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0]
---------ref---------->
[ 1/2    0    0  1/2    0    0    0    0 -1/2    0    0  1/2    0    0    0    0]
[   0    1    0    0    0    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    1    0    0    0    0    0    0    0    0    0    0    0    0    0]
[ 1/2    0    0  1/2    0    0    0    0  1/2    0    0 -1/2    0    0    0    0]
[   0    0    0    0    1    0    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    1    0    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    1    0    0    0    0    0    0    0    0    0]
[   0    0    0    0    0    0    0    1    0    0    0    0    0    0    0    0]
[-1/2    0    0  1/2    0    0    0    0  1/2    0    0  1/2    0    0    0    0]
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************************
[2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2]
---------ref---------->
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[ 1/2    0    0 -1/2    0    0    0    0    0    0    0    0  1/2    0    0  1/2]
************************
[2, 0, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, 2, 0, 0]
---------ref---------->
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************************
[2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0]
---------ref---------->
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************************
[2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0]
---------ref---------->
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************************
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
---------ref---------->
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************************