The realization space is
  [1   1   x1^2 - x1 - 1   0   0   1           1    x1^3 - x1^2 - x1          0       x1           1]
  [0   1              -1   1   0   0           1                 -x1          1        1          x1]
  [0   0               0   0   1   1   x1^2 - x1   x1^3 - 2*x1^2 + 1   x1^2 - 1   x1 - 1   x1^2 - x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (-x1^8 + 3*x1^7 - x1^6 - 5*x1^5 + 5*x1^4 + 2*x1^3 - 4*x1^2 + 1)
avoiding the zero loci of the polynomials
RingElem[x1 - 1, x1 + 1, x1, x1^2 - x1 - 1, x1^4 - x1^3 - x1^2 + x1 + 1, x1^3 - x1^2 + 1, 2*x1^2 - 1, x1^4 - x1 - 1, 2*x1 + 1]