No need for finding / calculating any phi or angle here. Just simpler than it is written.
Find a subset that divides the vector into equal halves
Generate a random matrix A of (1,-1)
Find the nearest integer
Is this matrix orthogonal?
Create a square matrix of multiples
Solve the set of simultaneous linear equations
Pythagorean perfect squares: find the square of the hypotenuse and the length of the other side
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