The realization space is
  [1   1     -x1^2   0   0   1           x1 - 1                 -x1^2        0               1           x1 - 1]
  [0   1   -x1 + 1   1   0   0           x1 - 1               -x1 + 1   x1 - 1      -x1^2 + x1        x1^2 - x1]
  [0   0         0   0   1   1   -x1^2 + x1 - 1   -x1^4 + x1^3 - x1^2      -x1   x1^2 - x1 + 1   -x1^2 + x1 - 1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^10 - 3*x1^9 + 4*x1^8 - 3*x1^7 + x1^6)
avoiding the zero loci of the polynomials
RingElem[x1, x1^2 - x1 + 1, x1 - 1, x1 + 1, x1^3 - x1^2 + 1, x1^4 - 2*x1^3 + x1 - 1, x1^3 - x1 + 1, x1^3 - x1^2 + x1 + 1, x1^2 - x1 - 1, x1^2 + 1]