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