I1  [pi,pi] = 0 in coordinates, symbolic lambda
  ok    I1 n=3  1 triples, 0 nonzero
  ok    I1 n=4  4 triples, 0 nonzero
  ok    I1 n=5  10 triples, 0 nonzero
  ok    I1 n=6  20 triples, 0 nonzero
I2  Jacobi identity for arbitrary functions f, g, h (n = 3)
  ok    I2  Jacobiator expanded (90 terms in one cyclic summand)
  ok    I2 CONTROL non-Poisson quadratic bivector: Jacobiator nonzero  
I3  {x^alpha, x^beta} = lambda(alpha,beta) x^(alpha+beta), symbolic exponents
  ok    I3 n=3  
  ok    I3 n=4  
I4  Lemma 2.3(b)
  ok    I4a lambda k = 0  
  ok    I4b lambda(alpha,beta) = k.(alpha x beta)  
I5  Remark 6.1: {f,g} dx^dy^dz = df^dg^kappa (arbitrary f, g)
  ok    I5  
  ok    I5 CONTROL with b and c exchanged in kappa: identity fails  
I6  Proposition 6.2 with an arbitrary function phi
  ok    I6a S(x,y,z) = 0  (a1,a2,a3) = (b*y*z, c*x*z, a*x*y)
  ok    I6b [phi pi0, phi pi0] = 0 (coordinates)  
  ok    I6c Jacobi for phi*{,}_0 on arbitrary f,g,h  
  ok    I6d CONTROL n=4, phi = x1 + x2: not Poisson (must have nonzero triples)  2 of 4 triples nonzero
I7  Lemma 2.1(b): rank(D lambda D) = rank(lambda) on the torus
  ok    I7  100/100 random rational lambda and torus points
I8  constants of Theorems A and D
  ok    I8a Theorem A: lambda k = 0 with k = (sqrt2, sqrt3, 1)  
  ok    I8b minimal polynomial of sqrt2+sqrt3  t**4 - 10*t**2 + 1
  ok    I8c Pf(lambda) = theta^5  theta**5
  ok    I8d det(lambda) = theta^10  
  ok    I8e det(D lambda D) = theta^10 (x1x2x3x4)^2  
  ok    I8f minimal polynomial of 2^(1/6)  t**6 - 2
  ok    I8g X^6 - 2 irreducible over Q (factor_list)  
I9  Example 4.1: kernel vectors and relation vectors
  ok    I9 (a)  (a,b,c) = (1, 2, 3), k = (2, 3, 1), nu = None
  ok    I9 (b)  (a,b,c) = (1, sqrt(2), -sqrt(2) - 1), k = (sqrt(2), -sqrt(2) - 1, 1), nu = (1, 1, 1)
  ok    I9 (c)  (a,b,c) = (1, 1, sqrt(2)), k = (1, sqrt(2), 1), nu = (1, 0, -1)
  ok    I9 (d)  (a,b,c) = (sqrt(2), -2, 1), k = (-2, 1, sqrt(2)), nu = (1, 2, 0)
TOTAL failures: 0
running time: 1 s
