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