Use R::=QQ[x[1..6]];
G:=[-x[4] - x[6], -x[1], x[2]^2 + x[3]^2,
-4x[4]^2 - 5/2x[5]^2 + 2x[4]x[6] - 4x[6]^2,
-2x[2]x[3]x[4] + x[2]^2x[5] - x[3]^2x[5] + 2x[2]x[3]x[6],
-x[2]^2x[4] + x[3]^2x[4] - 2x[2]x[3]x[5] + x[2]^2x[6] - x[3]^2x[6]];
SB.Sagbi(G);
SB.ReducedSagbi(G);
-------------------------------------------------------
-- output:
-- a SAGBI-basis of K[G]
[
x[4] + x[6],
x[1],
x[2]^2 + x[3]^2,
x[4]^2 + 5/8x[5]^2 - 1/2x[4]x[6] + x[6]^2,
x[2]x[3]x[4] - 1/2x[2]^2x[5] + 1/2x[3]^2x[5] - x[2]x[3]x[6],
x[2]^2x[4] - x[3]^2x[4] + 2x[2]x[3]x[5] - x[2]^2x[6] + x[3]^2x[6],
x[5]^2 - 4x[4]x[6],
x[3]^2x[4] - x[2]x[3]x[5] + x[2]^2x[6]]
-------------------------------
-- the reduced SAGBI-basis of K[G]
[
x[4] + x[6],
x[1],
x[2]^2 + x[3]^2,
x[2]x[3]x[4] - 1/2x[2]^2x[5] + 1/2x[3]^2x[5] - x[2]x[3]x[6],
x[5]^2 - 4x[4]x[6],
x[3]^2x[4] - x[2]x[3]x[5] + x[2]^2x[6]]
-------------------------------
-- Done.
-------------------------------
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