This applet computes products in the free algebra on 7 generators a, b, c, d, e, f, g with relations

gf = fg + bc - cb + de - ed
ge = eg + ac - ca - df + fd
gd = dg - ab + ba + ef - fe
gc = cg + ea - ae + fb - bf
gb = bg + ad - da + cf - fc
ga = ag + db - bd + ce - ec
fa = af + eb - be + dc - cd.

The diamond lemma tells us that a basis for this algebra is given by the set of all words (in the generators) that do not contain any one of

gf, ge, gd, gc, gb, ga, fa

as a subword. The result is given in this basis (with terms ordered lexicographically).