This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i55250U CPU at 1.60GHz.
There were 12 solutions found in 0.0562 seconds (with a Bezout bound of 16).
Reference: "Solving polynomial systems" by TienYien Li (pages 3339).
See also: http://homepages.math.uic.edu/~jan/Demo/lorentz.html.

The object lorentz is a method function.