cartan matrix :
   2  -1   0   0
  -1   2  -1   0
   0  -1   2  -1
   0   0  -1   2

root basis :
   2  -1   0   0
   3   0  -1   0
   4   0   0  -1
   5   0   0   0

coroot basis :
   1  -2   1   0
   0   1  -2   1
   0   0   1  -2
   0   0   0   1

positive roots :
[-1,-1,-1,0]
[-1,-1,0,0]
[-1,0,0,0]
[0,-1,-1,0]
[0,-1,0,0]
[0,0,-1,0]
[1,2,3,5]
[1,2,4,5]
[1,3,4,5]
[2,3,4,5]

positive coroots :
[-1,0,-1,1]
[-1,-1,1,0]
[-2,1,0,0]
[1,-1,-1,1]
[1,-2,1,0]
[0,1,-2,1]
[0,0,-1,1]
[0,-1,1,0]
[-1,1,0,0]
[1,0,0,0]