Roots of E8 ?

{alpha[1]: 1,
 alpha[1] + alpha[2] + alpha[3] + alpha[4]: 1,
 alpha[1] + alpha[2] + alpha[3] + alpha[4] + alpha[5]: 1,
 alpha[1] + alpha[3]: 1,
 alpha[1] + alpha[3] + alpha[4]: 1,
 alpha[1] + alpha[3] + alpha[4] + alpha[5]: 1,
 alpha[1] + alpha[3] + alpha[4] + alpha[5] + alpha[6]: 1,
 alpha[2]: 1,
 alpha[2] + alpha[3] + alpha[4]: 1,
 alpha[2] + alpha[3] + alpha[4] + alpha[5]: 1,
 alpha[2] + alpha[3] + alpha[4] + alpha[5] + alpha[6]: 1,
 alpha[2] + alpha[3] + 2*alpha[4] + alpha[5]: 1,
 alpha[2] + alpha[4]: 1,
 alpha[2] + alpha[4] + alpha[5]: 1,
 alpha[2] + alpha[4] + alpha[5] + alpha[6]: 1,
 alpha[2] + alpha[4] + alpha[5] + alpha[6] + alpha[7]: 1,
 alpha[3]: 1,
 alpha[3] + alpha[4]: 1,
 alpha[3] + alpha[4] + alpha[5]: 1,
 alpha[3] + alpha[4] + alpha[5] + alpha[6]: 1,
 alpha[3] + alpha[4] + alpha[5] + alpha[6] + alpha[7]: 1,
 alpha[4]: 1,
 alpha[4] + alpha[5]: 1,
 alpha[4] + alpha[5] + alpha[6]: 1,
 alpha[4] + alpha[5] + alpha[6] + alpha[7]: 1,
 alpha[4] + alpha[5] + alpha[6] + alpha[7] + alpha[8]: 1,
 alpha[5]: 1,
 alpha[5] + alpha[6]: 1,
 alpha[5] + alpha[6] + alpha[7]: 1,
 alpha[5] + alpha[6] + alpha[7] + alpha[8]: 1,
 alpha[6]: 1,
 alpha[6] + alpha[7]: 1,
 alpha[6] + alpha[7] + alpha[8]: 1,
 alpha[7]: 1,
 alpha[7] + alpha[8]: 1,
 alpha[8]: 1}