This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 12 solutions found in 0.0562 seconds (with a Bezout bound of 16).
Reference: "Solving polynomial systems" by Tien-Yien Li (pages 33-39).
See also: http://homepages.math.uic.edu/~jan/Demo/lorentz.html.
i1 : lorentz(QQ)
o1 = {x x - x x - x + 1, x x - x x - x + 1, - x x + x x - x + 1,
1 2 1 3 4 2 3 2 4 1 1 3 3 4 2
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x x - x x - x + 1}
1 4 2 4 3
o1 : List
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The object lorentz is a method function.