known_4i := [ 

P(3,3,3,3) = 6*P(3,3)^2 +P(3,3)*mu[1]^2 -3*P(2,2) +2*mu[1]*P(2,3) -4*
   P(3,3)*mu[2] -2*mu[4],

P(2,3,3,3) = 6*P(2,3)*P(3,3) +P(2,3)*mu[1]^2 +3*P(3,3)*mu[3]
   -P(2,3)*mu[2]  -mu[5] -mu[1]*P(2,2),

P(2,2,3,3) = 2*P(3,3)*P(2,2) +P(3,3)*mu[3]*mu[1] +4*P(2,3)^2
   -P(2,2)*mu[2] +2*mu[6]  +3*P(2,3)*mu[3] +P(2,3)*mu[1]*mu[2] +4*P(1, 3),

P(2,2,2,3) = 6*P(2,2)*P(2,3) + 4*mu[1]*P(1,3) +mu[3]*mu[1]*P(2,3)
   +P(3,3)*mu[2]*mu[3] +2*mu[3]*mu[4] +P(2,3)*mu[2]^2 +4*mu[4]*P(2,3)
   +3*mu[3]*P(2,2) +2*mu[6]*mu[1] +mu[5]*mu[2] -2*P(3,3)*mu[5],

P(2,2,2,2) = -2*P(2,3)*mu[2]*mu[3] +mu[2]*mu[1]*mu[5]
   +2*mu[1]*mu[4]*mu[3] +24*P(1,3)*P(3,3) +4*P(1,3)*mu[1]^2
   -4*P(1,3)*mu[2] -4*P(1,3,3,3) +4*P(2,3)*mu[5] +2*mu[1]^2*mu[6]
   -2*mu[2]*mu[6] +mu[3]*mu[5] -3*P(3,3)*mu[3]^2 +12*P(3,3)*mu[6]
   +6*P(2,2)^2 + 4*mu[4]*P(2,2) +P(2,2)*mu[2]^2 +4*mu[3]*mu[1]*P(2,2),

P(1,2,3,3) = -2*P(3,3)*mu[1]*P(1,3) +2*P(3,3)*P(1,2)
   -1/3*P(1,3)*mu[1]^3 +1/3 *mu[1]*P(1,3,3,3) +1/3*P(1,2)*mu[1]^2
   +4*P(1,3)*P(2,3) +3*P(1,3)*mu[3] +1/3*mu[8]*mu[1]
   +4/3*P(1,3)*mu[1]*mu[2] -P(1,2)*mu[2] +mu[9],

P(1,2,2,3) = -2*P(3,3)*P(1 ,3)*mu[2] -2*P(3,3)*mu[8] -2/3*mu[8]*mu[2]
   +1/3*P(1,3,3,3)*mu[2] +3*mu[3]*P(1,2) +4 *P(2,3)*P(1,2)
   +4*mu[4]*P(1,3) +4/3*mu[2]^2*P(1,3) -2*P(1,1)
   -1/3*P(1,3)*mu[1]^2*mu[2] +1/3*mu[1]*P(1,2)*mu[2] +2*P(1,3)*P(2,2)
   +mu[3]*mu[1]*P(1,3),

P(1,2,2,2) = 6*P(3,3)*mu[9] -P(1,3,3,3)*mu[3] +4*P(1,3)*mu[5]
   +P(1,2)*mu[2]^2 -mu[2]*mu[9] +4*mu [4]*P(1,2) -2*mu[1]*P(1,1)
   +6*P(2,2)*P(1,2) +6*P(1,3)*P(3,3)*mu[3] -3*P(1,3)*mu[2] *mu[3]
   +P(1,3)*mu[1]^2*mu[3] +3*mu[1]*mu[3]*P(1,2) -mu[8]*mu[1]*mu[2],

P(1,1,3,3) = 4*P(1,3)^2 +2*P(3,3)*P(1,1) -P(2,3)*mu[9] +2*P(1,3)*mu[6]
   +mu[8]*P(2,2) -P(1,2)* mu[5] +2/3*P(1,3,3,3)*mu[4] +2/3*mu[4]*mu[8]
   +2*P(3,3)*mu[8]*mu[2] -4*P(1,3)*P(3,3 )*mu[4]
   +2/3*P(1,3)*mu[4]*mu[2] +mu[1]*P(3,3)*mu[9] -mu[8]*mu[1]*P(2,3)
   +P(1,3)*mu [1]*mu[5] -2/3*P(1,3)*mu[1]^2*mu[4]
   +2/3*mu[1]*mu[4]*P(1,2),

P(1,1,2,3) = 2*P(1, 3)*mu[3]*mu[4] -mu[8]*P(3,3)*mu[3]
   -2*P(1,3)*P(3,3)*mu[5] +P(2,3)*mu[8]*mu[2] +4/3 *P(1,3)*mu[5]*mu[2]
   -mu[9]*P(2,2) +2*P(1,2)*mu[6] +1/3*P(1,3,3,3)*mu[5] +1/3*mu[8] *mu[5]
   +mu[9]*mu[1]*P(2,3) +4*P(1,2)*P(1,3) +2*P(2,3)*P(1,1)
   -1/3*P(1,3)*mu[1]^2* mu[5] +1/3*mu[1]*P(1,2)*mu[5],

P(1,1,2,2) = 2*mu[4]*mu[3]*P(1,2) +2*P(1,1)*P(2,2)
   +2/3*mu[1]^2*P(1,3)*mu[6] +4/3*mu[1]*P(1,2)*mu[6]
   +mu[9]*P(3,3)*mu[3] +mu[2]*P(2 ,3)*mu[9] +8*P(3,3)*mu[12]
   -2/3*P(1,3,3,3)*mu[6] +4*mu[8]*P(1,3) -2/3*mu[8]*mu[6] +
   4*P(3,3)*P(1,3)*mu[6] -mu[8]*P(2,3)*mu[3] +4*P(1,2)^2
   +P(1,3)*mu[3]*mu[5] -8/3*mu[ 2]*P(1,3)*mu[6] +mu[8]*mu[2]*P(2,2)
   +mu[2]*P(1,2)*mu[5],

P(1,1,1,3) = P(1,3)*mu[ 5]^2 +6*P(1,3)*mu[8]*mu[2]
   -2*mu[2]*P(3,3)*mu[12] -mu[1]^2*P(1,3)*mu[8] +4*mu[12]* mu[1]*P(2,3)
   +P(1,2)*mu[8]*mu[1] +P(3,3)*mu[5]*mu[9] -2*mu[4]*P(2,3)*mu[9] +mu[1]*
   P(1,3)*mu[9] -6*P(1,3)*P(3,3)*mu[8] -2*P(3,3)*mu[8]*mu[6]
   +6*P(1,3)*P(1,1) +mu[8]* P(1,3,3,3) -4*mu[12]*mu[4] +3*P(1,2)*mu[9]
   -6*mu[12]*P(2,2) -mu[8]*P(2,3)*mu[5] +4* mu[4]*P(1,3)*mu[6],

P(1,1,1,2) = 6*P(1,2)*P(1,1) +6*mu[12]*P(3,3)*mu[3] +3*mu[8]
   *P(1,3)*mu[3] -2*P(2,3)*mu[8]*mu[6] -mu[8]^2*mu[1]
   +5*P(1,2)*mu[8]*mu[2] +4*mu[12] *P(2,3)*mu[2] -2*mu[1]*mu[12]*P(2,2)
   +4*mu[4]*P(1,2)*mu[6] -P(2,2)*mu[8]*mu[5] +P( 1,2)*mu[5]^2
   +4*mu[12]*mu[5] -P(1,3,3,3)*mu[9] -4*mu[1]*mu[12]*mu[4] 
   +mu[1]^2*P(1,3)*mu[9] +3*mu[1]*P(1,2)*mu[9] -2*mu[4]*mu[9]*P(2,2)
   +mu[5]*P(2,3)*mu[9] -4*mu[2]* mu[9]*P(1,3) +6*mu[9]*P(1,3)*P(3,3)
   -3*mu[8]*mu[9],

P(1,1,1,1) = 4*mu[9]*mu[4]*P (1,2) -8*mu[12]*mu[4]^2
   -2*mu[2]^2*mu[12]*mu[4] -3*mu[8]^2*P(2,2) -2*mu[4]*mu[8]^2
   +mu[5]^2*P(1,1) +6*P(1,1)^2 -3*P(3,3)*mu[9]^2 -4*P(1,3,3,3)*mu[12]
   +24*mu[12]*P(1, 3)*P(3,3) +12*mu[12]*P(2,3)*mu[5]
   +mu[9]*mu[4]*mu[5]*mu[2] -6*mu[1]*mu[12]*mu[3]* mu[4]
   +mu[1]*mu[12]*mu[5]*mu[2] +2*mu[8]*mu[6]^2 +2*mu[8]^2*mu[2]^2
   -mu[9]*mu[5]* mu[6] -2*mu[9]*mu[5]*P(1,3) +4*mu[6]*P(1,1)*mu[4]
   +4*mu[8]*mu[6]*P(1,3) +8*mu[8]* mu[2]*P(1,1) -6*mu[2]*mu[12]*mu[6]
   -12*mu[2]*mu[12]*P(1,3) +4*mu[1]^2*mu[12]*P(1, 3)
   +2*mu[1]^2*mu[12]*mu[6] +2*P(1,2)*mu[8]*mu[5] -6*mu[8]*P(2,3)*mu[9]
   -12*P(2,2)* mu[12]*mu[4] +mu[5]^2*mu[8]*mu[2]
   +2*mu[6]*mu[1]*mu[9]*mu[4] +mu[1]*mu[8]*mu[6]* mu[5]
   +12*mu[6]*P(3,3)*mu[12] +4*mu[9]*mu[1]*P(1,1) +2*mu[9]*mu[4]^2*mu[3]
   +9*mu[3 ]*mu[12]*mu[5] -2*mu[8]^2*mu[3]*mu[1] -6*mu[8]*mu[3]*mu[9]
   +2*mu[2]*mu[9]*mu[8]* mu[1] +mu[3]*mu[4]*mu[8]*mu[5]
   +2*mu[6]*mu[8]*mu[2]*mu[4] +2*mu[2]*mu[9]^2]];