Issue with liealgebras\weyl_groups

[pb-elie]

I am investigating the Lie algebra E8, and found the sympy package liealgebras very useful.

I have however an issue with the matrix_form of the Weyl reflection corresponding to the first root:

from sympy.liealgebras.weyl_group import WeylGroup

wg = WeylGroup("E8")
m = wg.matrix_form('r1'))

1. m is not symmetric;
2. m**2 is not equal to the identity;

To see where the problem is I made code for the reflection matrix directly from the root (at least for E8);

def reflection_matrix(root):
    v = Matrix(root)
    vt = Transpose(v)
    R = eye(8) - v * vt
    return R  

For the first root:

root =  [1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2]

the matrix of the corresponding Weyl reflection is:

[3/4   1/4   1/4   1/4   1/4   1/4   1/4   -1/4]
[                                              ]
[1/4   3/4   -1/4  -1/4  -1/4  -1/4  -1/4  1/4 ]
[                                              ]
[1/4   -1/4  3/4   -1/4  -1/4  -1/4  -1/4  1/4 ]
[                                              ]
[1/4   -1/4  -1/4  3/4   -1/4  -1/4  -1/4  1/4 ]
[                                              ]
[1/4   -1/4  -1/4  -1/4  3/4   -1/4  -1/4  1/4 ]
[                                              ]
[1/4   -1/4  -1/4  -1/4  -1/4  3/4   -1/4  1/4 ]
[                                              ]
[1/4   -1/4  -1/4  -1/4  -1/4  -1/4  3/4   1/4 ]
[                                              ]
[-1/4  1/4   1/4   1/4   1/4   1/4   1/4   3/4 ]

This is symmetric, has 3/4 on the diagonal and its square is the identity, so I think this one is correct.
It is different from the one in weyl_groups --> matrix_form.

I only looked at E8 so I don't know about the other groups.

I am using Python 3.8.5 and Sympy 1.11.1