Example 3 is taken from the book "Ideals, Varieties, and Algorithms",
(example 2, p. 93-94), and it is a commutative example.

First, we set up the problem and find a Grobner basis w.r.t. the length
left-lexicographic ordering implicitly assumed in {\GBNP}:

gap> LoadPackage("gbnp");
─────────────────────────────────────────────────────────────────────────────
Loading  GBNP 1.0.2 (Non-commutative Gröbner bases)
by A.M. Cohen (http://www.win.tue.nl/~amc) and
   J.W. Knopper (J.W.Knopper@tue.nl).
Homepage: http://mathdox.org/products/gbnp/
─────────────────────────────────────────────────────────────────────────────
true
gap> A := FreeAssociativeAlgebraWithOne(Rationals,"x","y","z");;
gap> x := A.x;; y := A.y;; z := A.z;; id := One(A);;
gap> polys := [ x^2 + y^2 + z^2 - id, x^2 + z^2 - y, x-z,
>            x*y-y*x, x*z-z*x, y*z-z*y];;
gap> reps := GP2NPList(polys);;
gap> gb := Grobner(reps);;
gap> NP2GPList(gb,A);
[ (1)*z+(-1)*x, (1)*x^2+(-1/2)*y, (1)*y*x+(-1)*x*y, 
  (1)*y^2+(2)*x^2+(-1)*<identity ...> ]

gap> ml := NCMonomialLeftLexicographicOrdering(A,[3,2,1]);
NCMonomialLeftLexicographicOrdering([ (1)*z, (1)*y, (1)*x ])
gap> PatchGBNP(ml);
LtNP patched.
GtNP patched.
gap> gb := Grobner(reps);;
gap> NP2GPList(gb,A);
[ (1)*z^4+(1/2)*z^2+(-1/4)*<identity ...>, (1)*y+(-2)*z^2, (1)*x+(-1)*z ]

