Part A: Jacobi identity on all coordinate triples (theta = 2^(1/6))
  Jacobiator == 0: True
Part B: Pfaffian and minimal polynomial
  Pf(lambda) = l12 l34 - l13 l24 + l14 l23 = 2**(5/6)  (= theta^5, nonzero)
  det A(x) = 2*2**(2/3)*x1**2*x2**2*x3**2*x4**2   (= Pf(lambda)^2 (x1 x2 x3 x4)^2)
  minimal polynomial of theta over Q: x**6 - 2  degree 6
  => 1, theta, ..., theta^5 are linearly independent over Q (no rational isotropic plane).
Part C: commuting monomial pairs with linearly independent exponents, degree <= 4
  theta-bracket: 0 pairs (expected 0) []
  control, all lambda_ij = 1: 156 pairs, e.g. [((0, 0, 1, 0), (0, 1, 0, 1)), ((0, 0, 1, 0), (0, 1, 1, 1)), ((0, 0, 1, 0), (0, 1, 2, 1))]
Part D: exact centralizer dimensions (rank over Q(theta))
  f = x1+x2+x3+x4, deg <= 4: space dim 70, centralizer dim 5 (polynomials in f of degree <= 4: 5)   [3.2s]
  f = x1x2+x3+x4^2, deg <= 4: space dim 70, centralizer dim 3 (polynomials in f of degree <= 4: 3)   [4.6s]
Part E: Lemma 4.2 on random commuting pairs (graded lex order)
  lambda(LE(p), LE(q)) == 0 for all tested commuting pairs: True
total time 57.2s
exit 0
