Simpler code.
RootReduce[ With[{a = GoldenRatio}, ResourceFunction["SignedPermutations"][#, "Even"] & /@ { {0, 0, 2, 2}, {a, a, a, 1/a^2}, {1, 1, 1, Sqrt[5]}, {1/a, 1/a, 1/a, a^2}, {0, 1/a, a, Sqrt[5]}, {0, 1/a^2, 1, a^2}, {1/a, 1, a, 2}}]]
Ah, nice!
Truly awesome, thx Clayton Shonkwiler.
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