#2457L1. It has purely coclosed G2-structure


D.<e1,e2,e3,e4,e5,e7,f1,f2,f3,f4,f5,f7,g1,g2,g3,g4,g5,g7,h1,h2,h3,h4,h5,h7> = GradedCommutativeAlgebra(QQ)
N=D.cdg_algebra({})
N.inject_variables()
psie=-e1*e2*e3+e2*e3*e4+e1*e3*e5+e2*e4*e7-2*e3*e4*e5+e1*e5*e7
psif=-f1*f2*f3+f2*f3*f4+f1*f3*f5+f2*f4*f7-2*f3*f4*f5+f1*f5*f7
psig=-g1*g2*g3+g2*g3*g4+g1*g3*g5+g2*g4*g7-2*g3*g4*g5+g1*g5*g7
psih=-h1*h2*h3+h2*h3*h4+h1*h3*h5+h2*h4*h7-2*h3*h4*h5+h1*h5*h7
Cgfe=g1*f2*f3*e4*e5*e7+g1*f2*e3*f4*e5*e7+g1*f2*e3*e4*f5*e7+g1*f2*e3*e4*e5*f7+g1*e2*f3*f4*e5*e7+g1*e2*f3*e4*f5*e7+g1*e2*f3*e4*e5*f7+g1*e2*e3*f4*f5*e7+g1*e2*e3*f4*e5*f7+g1*e2*e3*e4*f5*f7+f1*g2*f3*e4*e5*e7+f1*g2*e3*f4*e5*e7+f1*g2*e3*e4*f5*e7+f1*g2*e3*e4*e5*f7+e1*g2*f3*f4*e5*e7+e1*g2*f3*e4*f5*e7+e1*g2*f3*e4*e5*f7+e1*g2*e3*f4*f5*e7+e1*g2*e3*f4*e5*f7+e1*g2*e3*e4*f5*f7+f1*f2*g3*e4*e5*e7+f1*e2*g3*f4*e5*e7+f1*e2*g3*e4*f5*e7+f1*e2*g3*e4*e5*f7+e1*f2*g3*f4*e5*e7+e1*f2*g3*e4*f5*e7+e1*f2*g3*e4*e5*f7+e1*e2*g3*f4*f5*e7+e1*e2*g3*f4*e5*f7+e1*e2*g3*e4*f5*f7+f1*f2*e3*g4*e5*e7+f1*e2*f3*g4*e5*e7+f1*e2*e3*g4*f5*e7+f1*e2*e3*g4*e5*f7+e1*f2*f3*g4*e5*e7+e1*f2*e3*g4*f5*e7+e1*f2*e3*g4*e5*f7+e1*e2*f3*g4*f5*e7+e1*e2*f3*g4*e5*f7+e1*e2*e3*g4*f5*f7+f1*f2*e3*e4*g5*e7+f1*e2*f3*e4*g5*e7+f1*e2*e3*f4*g5*e7+f1*e2*e3*e4*g5*f7+e1*f2*f3*e4*g5*e7+e1*f2*e3*f4*g5*e7+e1*f2*e3*e4*g5*f7+e1*e2*f3*f4*g5*e7+e1*e2*f3*e4*g5*f7+e1*e2*e3*f4*g5*f7+f1*f2*e3*e4*e5*g7+f1*e2*f3*e4*e5*g7+f1*e2*e3*f4*e5*g7+f1*e2*e3*e4*f5*g7+e1*f2*f3*e4*e5*g7+e1*f2*e3*f4*e5*g7+e1*f2*e3*e4*f5*g7+e1*e2*f3*f4*e5*g7+e1*e2*f3*e4*f5*g7+e1*e2*e3*f4*f5*g7
Cfgh=f1*g2*g3*h4*h5*h7+f1*g2*h3*g4*h5*h7+f1*g2*h3*h4*g5*h7+f1*g2*h3*h4*h5*g7+f1*h2*g3*g4*h5*h7+f1*h2*g3*h4*g5*h7+f1*h2*g3*h4*h5*g7+f1*h2*h3*g4*g5*h7+f1*h2*h3*g4*h5*g7+f1*h2*h3*h4*g5*g7+g1*f2*g3*h4*h5*h7+g1*f2*h3*g4*h5*h7+g1*f2*h3*h4*g5*h7+g1*f2*h3*h4*h5*g7+h1*f2*g3*g4*h5*h7+h1*f2*g3*h4*g5*h7+h1*f2*g3*h4*h5*g7+h1*f2*h3*g4*g5*h7+h1*f2*h3*g4*h5*g7+h1*f2*h3*h4*g5*g7+g1*g2*f3*h4*h5*h7+g1*h2*f3*g4*h5*h7+g1*h2*f3*h4*g5*h7+g1*h2*f3*h4*h5*g7+h1*g2*f3*g4*h5*h7+h1*g2*f3*h4*g5*h7+h1*g2*f3*h4*h5*g7+h1*h2*f3*g4*g5*h7+h1*h2*f3*g4*h5*g7+h1*h2*f3*h4*g5*g7+g1*g2*h3*f4*h5*h7+g1*h2*g3*f4*h5*h7+g1*h2*h3*f4*g5*h7+g1*h2*h3*f4*h5*g7+h1*g2*g3*f4*h5*h7+h1*g2*h3*f4*g5*h7+h1*g2*h3*f4*h5*g7+h1*h2*g3*f4*g5*h7+h1*h2*g3*f4*h5*g7+h1*h2*h3*f4*g5*g7+g1*g2*h3*h4*f5*h7+g1*h2*g3*h4*f5*h7+g1*h2*h3*g4*f5*h7+g1*h2*h3*h4*f5*g7+h1*g2*g3*h4*f5*h7+h1*g2*h3*g4*f5*h7+h1*g2*h3*h4*f5*g7+h1*h2*g3*g4*f5*h7+h1*h2*g3*h4*f5*g7+h1*h2*h3*g4*f5*g7+g1*g2*h3*h4*h5*f7+g1*h2*g3*h4*h5*f7+g1*h2*h3*g4*h5*f7+g1*h2*h3*h4*g5*f7+h1*g2*g3*h4*h5*f7+h1*g2*h3*g4*h5*f7+h1*g2*h3*h4*g5*f7+h1*h2*g3*g4*h5*f7+h1*h2*g3*h4*g5*f7+h1*h2*h3*g4*g5*f7
(-1/6)*psie*Cgfe*psif*psih*Cfgh*psig



D.<e1,e2,e3,e4,e5,e7,f1,f2,f3,f4,f5,f7,g1,g2,g3,g4,g5,g7,h1,h2,h3,h4,h5,h7,x1,x2,x3,x4,x5,x7,y1,y2,y3,y4,y5,y7,z1,z2,z3,z4,z5,z7> = GradedCommutativeAlgebra(QQ)
N=D.cdg_algebra({})
N.inject_variables()
psie=-e1*e2*e3+e2*e3*e4+e1*e3*e5+e2*e4*e7-2*e3*e4*e5+e1*e5*e7
psif=-f1*f2*f3+f2*f3*f4+f1*f3*f5+f2*f4*f7-2*f3*f4*f5+f1*f5*f7
psig=-g1*g2*g3+g2*g3*g4+g1*g3*g5+g2*g4*g7-2*g3*g4*g5+g1*g5*g7
psix=-x1*x2*x3+x2*x3*x4+x1*x3*x5+x2*x4*x7-2*x3*x4*x5+x1*x5*x7
psiy=-y1*y2*y3+y2*y3*y4+y1*y3*y5+y2*y4*y7-2*y3*y4*y5+y1*y5*y7
psiz=-z1*z2*z3+z2*z3*z4+z1*z3*z5+z2*z4*z7-2*z3*z4*z5+z1*z5*z7
psixyz=-x1*y2*z3+x2*y3*z4+x1*y3*z5+x2*y4*z7-2*x3*y4*z5+x1*y5*z7
Chxe=h1*x2*x3*e4*e5*e7+h1*x2*e3*x4*e5*e7+h1*x2*e3*e4*x5*e7+h1*x2*e3*e4*e5*x7+h1*e2*x3*x4*e5*e7+h1*e2*x3*e4*x5*e7+h1*e2*x3*e4*e5*x7+h1*e2*e3*x4*x5*e7+h1*e2*e3*x4*e5*x7+h1*e2*e3*e4*x5*x7+x1*h2*x3*e4*e5*e7+x1*h2*e3*x4*e5*e7+x1*h2*e3*e4*x5*e7+x1*h2*e3*e4*e5*x7+e1*h2*x3*x4*e5*e7+e1*h2*x3*e4*x5*e7+e1*h2*x3*e4*e5*x7+e1*h2*e3*x4*x5*e7+e1*h2*e3*x4*e5*x7+e1*h2*e3*e4*x5*x7+x1*x2*h3*e4*e5*e7+x1*e2*h3*x4*e5*e7+x1*e2*h3*e4*x5*e7+x1*e2*h3*e4*e5*x7+e1*x2*h3*x4*e5*e7+e1*x2*h3*e4*x5*e7+e1*x2*h3*e4*e5*x7+e1*e2*h3*x4*x5*e7+e1*e2*h3*x4*e5*x7+e1*e2*h3*e4*x5*x7+x1*x2*e3*h4*e5*e7+x1*e2*x3*h4*e5*e7+x1*e2*e3*h4*x5*e7+x1*e2*e3*h4*e5*x7+e1*x2*x3*h4*e5*e7+e1*x2*e3*h4*x5*e7+e1*x2*e3*h4*e5*x7+e1*e2*x3*h4*x5*e7+e1*e2*x3*h4*e5*x7+e1*e2*e3*h4*x5*x7+x1*x2*e3*e4*h5*e7+x1*e2*x3*e4*h5*e7+x1*e2*e3*x4*h5*e7+x1*e2*e3*e4*h5*x7+e1*x2*x3*e4*h5*e7+e1*x2*e3*x4*h5*e7+e1*x2*e3*e4*h5*x7+e1*e2*x3*x4*h5*e7+e1*e2*x3*e4*h5*x7+e1*e2*e3*x4*h5*x7+x1*x2*e3*e4*e5*h7+x1*e2*x3*e4*e5*h7+x1*e2*e3*x4*e5*h7+x1*e2*e3*e4*x5*h7+e1*x2*x3*e4*e5*h7+e1*x2*e3*x4*e5*h7+e1*x2*e3*e4*x5*h7+e1*e2*x3*x4*e5*h7+e1*e2*x3*e4*x5*h7+e1*e2*e3*x4*x5*h7
Chyf=h1*y2*y3*f4*f5*f7+h1*y2*f3*y4*f5*f7+h1*y2*f3*f4*y5*f7+h1*y2*f3*f4*f5*y7+h1*f2*y3*y4*f5*f7+h1*f2*y3*f4*y5*f7+h1*f2*y3*f4*f5*y7+h1*f2*f3*y4*y5*f7+h1*f2*f3*y4*f5*y7+h1*f2*f3*f4*y5*y7+y1*h2*y3*f4*f5*f7+y1*h2*f3*y4*f5*f7+y1*h2*f3*f4*y5*f7+y1*h2*f3*f4*f5*y7+f1*h2*y3*y4*f5*f7+f1*h2*y3*f4*y5*f7+f1*h2*y3*f4*f5*y7+f1*h2*f3*y4*y5*f7+f1*h2*f3*y4*f5*y7+f1*h2*f3*f4*y5*y7+y1*y2*h3*f4*f5*f7+y1*f2*h3*y4*f5*f7+y1*f2*h3*f4*y5*f7+y1*f2*h3*f4*f5*y7+f1*y2*h3*y4*f5*f7+f1*y2*h3*f4*y5*f7+f1*y2*h3*f4*f5*y7+f1*f2*h3*y4*y5*f7+f1*f2*h3*y4*f5*y7+f1*f2*h3*f4*y5*y7+y1*y2*f3*h4*f5*f7+y1*f2*y3*h4*f5*f7+y1*f2*f3*h4*y5*f7+y1*f2*f3*h4*f5*y7+f1*y2*y3*h4*f5*f7+f1*y2*f3*h4*y5*f7+f1*y2*f3*h4*f5*y7+f1*f2*y3*h4*y5*f7+f1*f2*y3*h4*f5*y7+f1*f2*f3*h4*y5*y7+y1*y2*f3*f4*h5*f7+y1*f2*y3*f4*h5*f7+y1*f2*f3*y4*h5*f7+y1*f2*f3*f4*h5*y7+f1*y2*y3*f4*h5*f7+f1*y2*f3*y4*h5*f7+f1*y2*f3*f4*h5*y7+f1*f2*y3*y4*h5*f7+f1*f2*y3*f4*h5*y7+f1*f2*f3*y4*h5*y7+y1*y2*f3*f4*f5*h7+y1*f2*y3*f4*f5*h7+y1*f2*f3*y4*f5*h7+y1*f2*f3*f4*y5*h7+f1*y2*y3*f4*f5*h7+f1*y2*f3*y4*f5*h7+f1*y2*f3*f4*y5*h7+f1*f2*y3*y4*f5*h7+f1*f2*y3*f4*y5*h7+f1*f2*f3*y4*y5*h7
Chzg=h1*z2*z3*g4*g5*g7+h1*z2*g3*z4*g5*g7+h1*z2*g3*g4*z5*g7+h1*z2*g3*g4*g5*z7+h1*g2*z3*z4*g5*g7+h1*g2*z3*g4*z5*g7+h1*g2*z3*g4*g5*z7+h1*g2*g3*z4*z5*g7+h1*g2*g3*z4*g5*z7+h1*g2*g3*g4*z5*z7+z1*h2*z3*g4*g5*g7+z1*h2*g3*z4*g5*g7+z1*h2*g3*g4*z5*g7+z1*h2*g3*g4*g5*z7+g1*h2*z3*z4*g5*g7+g1*h2*z3*g4*z5*g7+g1*h2*z3*g4*g5*z7+g1*h2*g3*z4*z5*g7+g1*h2*g3*z4*g5*z7+g1*h2*g3*g4*z5*z7+z1*z2*h3*g4*g5*g7+z1*g2*h3*z4*g5*g7+z1*g2*h3*g4*z5*g7+z1*g2*h3*g4*g5*z7+g1*z2*h3*z4*g5*g7+g1*z2*h3*g4*z5*g7+g1*z2*h3*g4*g5*z7+g1*g2*h3*z4*z5*g7+g1*g2*h3*z4*g5*z7+g1*g2*h3*g4*z5*z7+z1*z2*g3*h4*g5*g7+z1*g2*z3*h4*g5*g7+z1*g2*g3*h4*z5*g7+z1*g2*g3*h4*g5*z7+g1*z2*z3*h4*g5*g7+g1*z2*g3*h4*z5*g7+g1*z2*g3*h4*g5*z7+g1*g2*z3*h4*z5*g7+g1*g2*z3*h4*g5*z7+g1*g2*g3*h4*z5*z7+z1*z2*g3*g4*h5*g7+z1*g2*z3*g4*h5*g7+z1*g2*g3*z4*h5*g7+z1*g2*g3*g4*h5*z7+g1*z2*z3*g4*h5*g7+g1*z2*g3*z4*h5*g7+g1*z2*g3*g4*h5*z7+g1*g2*z3*z4*h5*g7+g1*g2*z3*g4*h5*z7+g1*g2*g3*z4*h5*z7+z1*z2*g3*g4*g5*h7+z1*g2*z3*g4*g5*h7+z1*g2*g3*z4*g5*h7+z1*g2*g3*g4*z5*h7+g1*z2*z3*g4*g5*h7+g1*z2*g3*z4*g5*h7+g1*z2*g3*g4*z5*h7+g1*g2*z3*z4*g5*h7+g1*g2*z3*g4*z5*h7+g1*g2*g3*z4*z5*h7
psie*Chxe*psix*psif*Chyf*psiy*psig*Chzg*psiz*psixyz


#psiplus=c*(x1*x2*x3+2*x2*x3*x4+2*x1*x3*x5+2*x3*x4*x5-x1*x2*x7-2*x4*x5*x7)


D.<e1,e2,e3,e4,e5,e7,f1,f2,f3,f4,f5,f7,x1,x2,x3,x4,x5,x7> = GradedCommutativeAlgebra(QQ,degrees=(1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2))
N=D.cdg_algebra({})
N.inject_variables()
psie=-e1*e2*e3+e2*e3*e4+e1*e3*e5+e2*e4*e7-2*e3*e4*e5+e1*e5*e7
psif=-f1*f2*f3+f2*f3*f4+f1*f3*f5+f2*f4*f7-2*f3*f4*f5+f1*f5*f7
omegaxf=-x1*f4+2*x2*f5+x3*f7-(-f1*x4+2*f2*x5+f3*x7)
Cxfe=x1*f2*f3*e4*e5*e7+x1*f2*e3*f4*e5*e7+x1*f2*e3*e4*f5*e7+x1*f2*e3*e4*e5*f7+x1*e2*f3*f4*e5*e7+x1*e2*f3*e4*f5*e7+x1*e2*f3*e4*e5*f7+x1*e2*e3*f4*f5*e7+x1*e2*e3*f4*e5*f7+x1*e2*e3*e4*f5*f7-x2*f1*f3*e4*e5*e7-x2*f1*e3*f4*e5*e7-x2*f1*e3*e4*f5*e7-x2*f1*e3*e4*e5*f7-x2*e1*f3*f4*e5*e7-x2*e1*f3*e4*f5*e7-x2*e1*f3*e4*e5*f7-x2*e1*e3*f4*f5*e7-x2*e1*e3*f4*e5*f7-x2*e1*e3*e4*f5*f7+x3*f1*f2*e4*e5*e7+x3*f1*e2*f4*e5*e7+x3*f1*e2*e4*f5*e7+x3*f1*e2*e4*e5*f7+x3*e1*f2*f4*e5*e7+x3*e1*f2*e4*f5*e7+x3*e1*f2*e4*e5*f7+x3*e1*e2*f4*f5*e7+x3*e1*e2*f4*e5*f7+x3*e1*e2*e4*f5*f7-x4*f1*f2*e3*e5*e7-x4*f1*e2*f3*e5*e7-x4*f1*e2*e3*f5*e7-x4*f1*e2*e3*e5*f7-x4*e1*f2*f3*e5*e7-x4*e1*f2*e3*f5*e7-x4*e1*f2*e3*e5*f7-x4*e1*e2*f3*f5*e7-x4*e1*e2*f3*e5*f7-x4*e1*e2*e3*f5*f7+x5*f1*f2*e3*e4*e7+x5*f1*e2*f3*e4*e7+x5*f1*e2*e3*f4*e7+x5*f1*e2*e3*e4*f7+x5*e1*f2*f3*e4*e7+x5*e1*f2*e3*f4*e7+x5*e1*f2*e3*e4*f7+x5*e1*e2*f3*f4*e7+x5*e1*e2*f3*e4*f7+x5*e1*e2*e3*f4*f7-x7*f1*f2*e3*e4*e5-x7*f1*e2*f3*e4*e5-x7*f1*e2*e3*f4*e5-x7*f1*e2*e3*e4*f5-x7*e1*f2*f3*e4*e5-x7*e1*f2*e3*f4*e5-x7*e1*f2*e3*e4*f5-x7*e1*e2*f3*f4*e5-x7*e1*e2*f3*e4*f5-x7*e1*e2*e3*f4*f5
1/2*psie*Cxfe*psif*omegaxf

# x1^2 + 2x2^2 + 3x3^2 - 2x3*x7 + 2x4^2 + 4x5^2 + x7^2
# [1,0,0,0,0,0],[0,2,0,0,0,0],[0,0,3,0,0,-1],[0,0,0,2,0,0],[0,0,0,0,4,0],[0,0,-1,0,0,1]
# The metric is positive definite 


A.<x1,x2,x3,x4,x5,x6,x7> = GradedCommutativeAlgebra(QQ)
M=A.cdg_algebra({x3: x1*x2, x4: x1*x3, x5:x2*x3, x6:x1*x4-x2*x5, x7:x1*x5+x2*x4})
M.inject_variables()
omega=-x1*x4+2*x2*x5+x3*x7
psi=-x1*x2*x3+x2*x3*x4+x1*x3*x5+x2*x4*x7-2*x3*x4*x5+x1*x5*x7
psiplus=x1*x2*x3+2*x2*x3*x4+2*x1*x3*x5+2*x3*x4*x5-x1*x2*x7-2*x4*x5*x7
eta=2/3*x6
omega*psi
psi*psiplus-(2/3)*omega^3
psi.differential()
omega*omega.differential()-psi*eta.differential()
omega^2*eta.differential()+2*psiplus*omega.differential(