#n15. It has purely coclosed G2-structure



D.<e1,e2,e3,e4,e5,e6,f1,f2,f3,f4,f5,f6,g1,g2,g3,g4,g5,g6,h1,h2,h3,h4,h5,h6> = GradedCommutativeAlgebra(QQ)
N=D.cdg_algebra({})
N.inject_variables()
psie=-e1*e2*e3+e1*e4*e6+e2*e4*e5+e2*e3*e6-e3*e5*e6
psif=-f1*f2*f3+f1*f4*f6+f2*f4*f5+f2*f3*f6-f3*f5*f6
psig=-g1*g2*g3+g1*g4*g6+g2*g4*g5+g2*g3*g6-g3*g5*g6
psih=-h1*h2*h3+h1*h4*h6+h2*h4*h5+h2*h3*h6-h3*h5*h6
Cgfe=g1*f2*f3*e4*e5*e6+g1*f2*e3*f4*e5*e6+g1*f2*e3*e4*f5*e6+g1*f2*e3*e4*e5*f6+g1*e2*f3*f4*e5*e6+g1*e2*f3*e4*f5*e6+g1*e2*f3*e4*e5*f6+g1*e2*e3*f4*f5*e6+g1*e2*e3*f4*e5*f6+g1*e2*e3*e4*f5*f6+f1*g2*f3*e4*e5*e6+f1*g2*e3*f4*e5*e6+f1*g2*e3*e4*f5*e6+f1*g2*e3*e4*e5*f6+e1*g2*f3*f4*e5*e6+e1*g2*f3*e4*f5*e6+e1*g2*f3*e4*e5*f6+e1*g2*e3*f4*f5*e6+e1*g2*e3*f4*e5*f6+e1*g2*e3*e4*f5*f6+f1*f2*g3*e4*e5*e6+f1*e2*g3*f4*e5*e6+f1*e2*g3*e4*f5*e6+f1*e2*g3*e4*e5*f6+e1*f2*g3*f4*e5*e6+e1*f2*g3*e4*f5*e6+e1*f2*g3*e4*e5*f6+e1*e2*g3*f4*f5*e6+e1*e2*g3*f4*e5*f6+e1*e2*g3*e4*f5*f6+f1*f2*e3*g4*e5*e6+f1*e2*f3*g4*e5*e6+f1*e2*e3*g4*f5*e6+f1*e2*e3*g4*e5*f6+e1*f2*f3*g4*e5*e6+e1*f2*e3*g4*f5*e6+e1*f2*e3*g4*e5*f6+e1*e2*f3*g4*f5*e6+e1*e2*f3*g4*e5*f6+e1*e2*e3*g4*f5*f6+f1*f2*e3*e4*g5*e6+f1*e2*f3*e4*g5*e6+f1*e2*e3*f4*g5*e6+f1*e2*e3*e4*g5*f6+e1*f2*f3*e4*g5*e6+e1*f2*e3*f4*g5*e6+e1*f2*e3*e4*g5*f6+e1*e2*f3*f4*g5*e6+e1*e2*f3*e4*g5*f6+e1*e2*e3*f4*g5*f6+f1*f2*e3*e4*e5*g6+f1*e2*f3*e4*e5*g6+f1*e2*e3*f4*e5*g6+f1*e2*e3*e4*f5*g6+e1*f2*f3*e4*e5*g6+e1*f2*e3*f4*e5*g6+e1*f2*e3*e4*f5*g6+e1*e2*f3*f4*e5*g6+e1*e2*f3*e4*f5*g6+e1*e2*e3*f4*f5*g6
Cfgh=f1*g2*g3*h4*h5*h6+f1*g2*h3*g4*h5*h6+f1*g2*h3*h4*g5*h6+f1*g2*h3*h4*h5*g6+f1*h2*g3*g4*h5*h6+f1*h2*g3*h4*g5*h6+f1*h2*g3*h4*h5*g6+f1*h2*h3*g4*g5*h6+f1*h2*h3*g4*h5*g6+f1*h2*h3*h4*g5*g6+g1*f2*g3*h4*h5*h6+g1*f2*h3*g4*h5*h6+g1*f2*h3*h4*g5*h6+g1*f2*h3*h4*h5*g6+h1*f2*g3*g4*h5*h6+h1*f2*g3*h4*g5*h6+h1*f2*g3*h4*h5*g6+h1*f2*h3*g4*g5*h6+h1*f2*h3*g4*h5*g6+h1*f2*h3*h4*g5*g6+g1*g2*f3*h4*h5*h6+g1*h2*f3*g4*h5*h6+g1*h2*f3*h4*g5*h6+g1*h2*f3*h4*h5*g6+h1*g2*f3*g4*h5*h6+h1*g2*f3*h4*g5*h6+h1*g2*f3*h4*h5*g6+h1*h2*f3*g4*g5*h6+h1*h2*f3*g4*h5*g6+h1*h2*f3*h4*g5*g6+g1*g2*h3*f4*h5*h6+g1*h2*g3*f4*h5*h6+g1*h2*h3*f4*g5*h6+g1*h2*h3*f4*h5*g6+h1*g2*g3*f4*h5*h6+h1*g2*h3*f4*g5*h6+h1*g2*h3*f4*h5*g6+h1*h2*g3*f4*g5*h6+h1*h2*g3*f4*h5*g6+h1*h2*h3*f4*g5*g6+g1*g2*h3*h4*f5*h6+g1*h2*g3*h4*f5*h6+g1*h2*h3*g4*f5*h6+g1*h2*h3*h4*f5*g6+h1*g2*g3*h4*f5*h6+h1*g2*h3*g4*f5*h6+h1*g2*h3*h4*f5*g6+h1*h2*g3*g4*f5*h6+h1*h2*g3*h4*f5*g6+h1*h2*h3*g4*f5*g6+g1*g2*h3*h4*h5*f6+g1*h2*g3*h4*h5*f6+g1*h2*h3*g4*h5*f6+g1*h2*h3*h4*g5*f6+h1*g2*g3*h4*h5*f6+h1*g2*h3*g4*h5*f6+h1*g2*h3*h4*g5*f6+h1*h2*g3*g4*h5*f6+h1*h2*g3*h4*g5*f6+h1*h2*h3*g4*g5*f6
(-1/6)*psie*Cgfe*psif*psih*Cfgh*psig



D.<e1,e2,e3,e4,e5,e6,f1,f2,f3,f4,f5,f6,g1,g2,g3,g4,g5,g6,h1,h2,h3,h4,h5,h6,x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,z1,z2,z3,z4,z5,z6> = GradedCommutativeAlgebra(QQ)
N=D.cdg_algebra({})
N.inject_variables()
psie=-e1*e2*e3+e1*e4*e6+e2*e4*e5+e2*e3*e6-e3*e5*e6
psif=-f1*f2*f3+f1*f4*f6+f2*f4*f5+f2*f3*f6-f3*f5*f6
psig=-g1*g2*g3+g1*g4*g6+g2*g4*g5+g2*g3*g6-g3*g5*g6
psix=-x1*x2*x3+x1*x4*x6+x2*x4*x5+x2*x3*x6-x3*x5*x6
psiy=-y1*y2*y3+y1*y4*y6+y2*y4*y5+y2*y3*y6-y3*y5*y6
psiz=-z1*z2*z3+z1*z4*z6+z2*z4*z5+z2*z3*z6-z3*z5*z6
psixyz=-x1*y2*z3+x1*y4*z6+x2*y4*z5+x2*y3*z6-x3*y5*z6
Chxe=h1*x2*x3*e4*e5*e6+h1*x2*e3*x4*e5*e6+h1*x2*e3*e4*x5*e6+h1*x2*e3*e4*e5*x6+h1*e2*x3*x4*e5*e6+h1*e2*x3*e4*x5*e6+h1*e2*x3*e4*e5*x6+h1*e2*e3*x4*x5*e6+h1*e2*e3*x4*e5*x6+h1*e2*e3*e4*x5*x6+x1*h2*x3*e4*e5*e6+x1*h2*e3*x4*e5*e6+x1*h2*e3*e4*x5*e6+x1*h2*e3*e4*e5*x6+e1*h2*x3*x4*e5*e6+e1*h2*x3*e4*x5*e6+e1*h2*x3*e4*e5*x6+e1*h2*e3*x4*x5*e6+e1*h2*e3*x4*e5*x6+e1*h2*e3*e4*x5*x6+x1*x2*h3*e4*e5*e6+x1*e2*h3*x4*e5*e6+x1*e2*h3*e4*x5*e6+x1*e2*h3*e4*e5*x6+e1*x2*h3*x4*e5*e6+e1*x2*h3*e4*x5*e6+e1*x2*h3*e4*e5*x6+e1*e2*h3*x4*x5*e6+e1*e2*h3*x4*e5*x6+e1*e2*h3*e4*x5*x6+x1*x2*e3*h4*e5*e6+x1*e2*x3*h4*e5*e6+x1*e2*e3*h4*x5*e6+x1*e2*e3*h4*e5*x6+e1*x2*x3*h4*e5*e6+e1*x2*e3*h4*x5*e6+e1*x2*e3*h4*e5*x6+e1*e2*x3*h4*x5*e6+e1*e2*x3*h4*e5*x6+e1*e2*e3*h4*x5*x6+x1*x2*e3*e4*h5*e6+x1*e2*x3*e4*h5*e6+x1*e2*e3*x4*h5*e6+x1*e2*e3*e4*h5*x6+e1*x2*x3*e4*h5*e6+e1*x2*e3*x4*h5*e6+e1*x2*e3*e4*h5*x6+e1*e2*x3*x4*h5*e6+e1*e2*x3*e4*h5*x6+e1*e2*e3*x4*h5*x6+x1*x2*e3*e4*e5*h6+x1*e2*x3*e4*e5*h6+x1*e2*e3*x4*e5*h6+x1*e2*e3*e4*x5*h6+e1*x2*x3*e4*e5*h6+e1*x2*e3*x4*e5*h6+e1*x2*e3*e4*x5*h6+e1*e2*x3*x4*e5*h6+e1*e2*x3*e4*x5*h6+e1*e2*e3*x4*x5*h6
Chyf=h1*y2*y3*f4*f5*f6+h1*y2*f3*y4*f5*f6+h1*y2*f3*f4*y5*f6+h1*y2*f3*f4*f5*y6+h1*f2*y3*y4*f5*f6+h1*f2*y3*f4*y5*f6+h1*f2*y3*f4*f5*y6+h1*f2*f3*y4*y5*f6+h1*f2*f3*y4*f5*y6+h1*f2*f3*f4*y5*y6+y1*h2*y3*f4*f5*f6+y1*h2*f3*y4*f5*f6+y1*h2*f3*f4*y5*f6+y1*h2*f3*f4*f5*y6+f1*h2*y3*y4*f5*f6+f1*h2*y3*f4*y5*f6+f1*h2*y3*f4*f5*y6+f1*h2*f3*y4*y5*f6+f1*h2*f3*y4*f5*y6+f1*h2*f3*f4*y5*y6+y1*y2*h3*f4*f5*f6+y1*f2*h3*y4*f5*f6+y1*f2*h3*f4*y5*f6+y1*f2*h3*f4*f5*y6+f1*y2*h3*y4*f5*f6+f1*y2*h3*f4*y5*f6+f1*y2*h3*f4*f5*y6+f1*f2*h3*y4*y5*f6+f1*f2*h3*y4*f5*y6+f1*f2*h3*f4*y5*y6+y1*y2*f3*h4*f5*f6+y1*f2*y3*h4*f5*f6+y1*f2*f3*h4*y5*f6+y1*f2*f3*h4*f5*y6+f1*y2*y3*h4*f5*f6+f1*y2*f3*h4*y5*f6+f1*y2*f3*h4*f5*y6+f1*f2*y3*h4*y5*f6+f1*f2*y3*h4*f5*y6+f1*f2*f3*h4*y5*y6+y1*y2*f3*f4*h5*f6+y1*f2*y3*f4*h5*f6+y1*f2*f3*y4*h5*f6+y1*f2*f3*f4*h5*y6+f1*y2*y3*f4*h5*f6+f1*y2*f3*y4*h5*f6+f1*y2*f3*f4*h5*y6+f1*f2*y3*y4*h5*f6+f1*f2*y3*f4*h5*y6+f1*f2*f3*y4*h5*y6+y1*y2*f3*f4*f5*h6+y1*f2*y3*f4*f5*h6+y1*f2*f3*y4*f5*h6+y1*f2*f3*f4*y5*h6+f1*y2*y3*f4*f5*h6+f1*y2*f3*y4*f5*h6+f1*y2*f3*f4*y5*h6+f1*f2*y3*y4*f5*h6+f1*f2*y3*f4*y5*h6+f1*f2*f3*y4*y5*h6
Chzg=h1*z2*z3*g4*g5*g6+h1*z2*g3*z4*g5*g6+h1*z2*g3*g4*z5*g6+h1*z2*g3*g4*g5*z6+h1*g2*z3*z4*g5*g6+h1*g2*z3*g4*z5*g6+h1*g2*z3*g4*g5*z6+h1*g2*g3*z4*z5*g6+h1*g2*g3*z4*g5*z6+h1*g2*g3*g4*z5*z6+z1*h2*z3*g4*g5*g6+z1*h2*g3*z4*g5*g6+z1*h2*g3*g4*z5*g6+z1*h2*g3*g4*g5*z6+g1*h2*z3*z4*g5*g6+g1*h2*z3*g4*z5*g6+g1*h2*z3*g4*g5*z6+g1*h2*g3*z4*z5*g6+g1*h2*g3*z4*g5*z6+g1*h2*g3*g4*z5*z6+z1*z2*h3*g4*g5*g6+z1*g2*h3*z4*g5*g6+z1*g2*h3*g4*z5*g6+z1*g2*h3*g4*g5*z6+g1*z2*h3*z4*g5*g6+g1*z2*h3*g4*z5*g6+g1*z2*h3*g4*g5*z6+g1*g2*h3*z4*z5*g6+g1*g2*h3*z4*g5*z6+g1*g2*h3*g4*z5*z6+z1*z2*g3*h4*g5*g6+z1*g2*z3*h4*g5*g6+z1*g2*g3*h4*z5*g6+z1*g2*g3*h4*g5*z6+g1*z2*z3*h4*g5*g6+g1*z2*g3*h4*z5*g6+g1*z2*g3*h4*g5*z6+g1*g2*z3*h4*z5*g6+g1*g2*z3*h4*g5*z6+g1*g2*g3*h4*z5*z6+z1*z2*g3*g4*h5*g6+z1*g2*z3*g4*h5*g6+z1*g2*g3*z4*h5*g6+z1*g2*g3*g4*h5*z6+g1*z2*z3*g4*h5*g6+g1*z2*g3*z4*h5*g6+g1*z2*g3*g4*h5*z6+g1*g2*z3*z4*h5*g6+g1*g2*z3*g4*h5*z6+g1*g2*g3*z4*h5*z6+z1*z2*g3*g4*g5*h6+z1*g2*z3*g4*g5*h6+z1*g2*g3*z4*g5*h6+z1*g2*g3*g4*z5*h6+g1*z2*z3*g4*g5*h6+g1*z2*g3*z4*g5*h6+g1*z2*g3*g4*z5*h6+g1*g2*z3*z4*g5*h6+g1*g2*z3*g4*z5*h6+g1*g2*g3*z4*z5*h6
psie*Chxe*psix*psif*Chyf*psiy*psig*Chzg*psiz*psixyz

#psiplus=c*(x1*x2*x4+x1*x3*x6+x2*x3*x5-x2*x4*x6+x4*x5*x6)


D.<e1,e2,e3,e4,e5,e6,f1,f2,f3,f4,f5,f6,x1,x2,x3,x4,x5,x6> = 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+e1*e4*e6+e2*e4*e5+e2*e3*e6-e3*e5*e6
psif=-f1*f2*f3+f1*f4*f6+f2*f4*f5+f2*f3*f6-f3*f5*f6
omegaxf=(x1*f2+x1*f5-x2*f6+x3*f4-1/2*(x1*f6-x2*f5))-(f1*x2+f1*x5-f2*x6+f3*x4-1/2*(f1*x6-f2*x5))
Cxfe=x1*f2*f3*e4*e5*e6+x1*f2*e3*f4*e5*e6+x1*f2*e3*e4*f5*e6+x1*f2*e3*e4*e5*f6+x1*e2*f3*f4*e5*e6+x1*e2*f3*e4*f5*e6+x1*e2*f3*e4*e5*f6+x1*e2*e3*f4*f5*e6+x1*e2*e3*f4*e5*f6+x1*e2*e3*e4*f5*f6-x2*f1*f3*e4*e5*e6-x2*f1*e3*f4*e5*e6-x2*f1*e3*e4*f5*e6-x2*f1*e3*e4*e5*f6-x2*e1*f3*f4*e5*e6-x2*e1*f3*e4*f5*e6-x2*e1*f3*e4*e5*f6-x2*e1*e3*f4*f5*e6-x2*e1*e3*f4*e5*f6-x2*e1*e3*e4*f5*f6+x3*f1*f2*e4*e5*e6+x3*f1*e2*f4*e5*e6+x3*f1*e2*e4*f5*e6+x3*f1*e2*e4*e5*f6+x3*e1*f2*f4*e5*e6+x3*e1*f2*e4*f5*e6+x3*e1*f2*e4*e5*f6+x3*e1*e2*f4*f5*e6+x3*e1*e2*f4*e5*f6+x3*e1*e2*e4*f5*f6-x4*f1*f2*e3*e5*e6-x4*f1*e2*f3*e5*e6-x4*f1*e2*e3*f5*e6-x4*f1*e2*e3*e5*f6-x4*e1*f2*f3*e5*e6-x4*e1*f2*e3*f5*e6-x4*e1*f2*e3*e5*f6-x4*e1*e2*f3*f5*e6-x4*e1*e2*f3*e5*f6-x4*e1*e2*e3*f5*f6+x5*f1*f2*e3*e4*e6+x5*f1*e2*f3*e4*e6+x5*f1*e2*e3*f4*e6+x5*f1*e2*e3*e4*f6+x5*e1*f2*f3*e4*e6+x5*e1*f2*e3*f4*e6+x5*e1*f2*e3*e4*f6+x5*e1*e2*f3*f4*e6+x5*e1*e2*f3*e4*f6+x5*e1*e2*e3*f4*f6-x6*f1*f2*e3*e4*e5-x6*f1*e2*f3*e4*e5-x6*f1*e2*e3*f4*e5-x6*f1*e2*e3*e4*f5-x6*e1*f2*f3*e4*e5-x6*e1*f2*e3*f4*e5-x6*e1*f2*e3*e4*f5-x6*e1*e2*f3*f4*e5-x6*e1*e2*f3*e4*f5-x6*e1*e2*e3*f4*f5
1/2*psie*Cxfe*psif*omegaxf

# x1^2 + x1*x2 + 2x2^2 + 2x2*x5 - x2*x6 + x3^2 + x4^2 + x5^2 - x5*x6 + x6^2
# [1,1/2,0,0,0,0],[1/2,2,0,0,1,-1/2],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,1,-1/2],[0,-1/2,0,0,-1/2,1]
# The metric is positive definite


A.<x1,x2,x3,x4,x5,x6,x7> = GradedCommutativeAlgebra(QQ)
M=A.cdg_algebra({ x4:x1*x2, x5:x1*x3, x6: x1*x4+x3*x5 })
M.inject_variables()
omega=x1*x2+x1*x5-x2*x6+x3*x4-1/2*x1*x6+1/2*x2*x5
psi=-x1*x2*x3+x1*x4*x6+x2*x4*x5+x2*x3*x6-x3*x5*x6
psiplus=-3/4*(x1*x2*x4+x1*x3*x6+x2*x3*x5-x2*x4*x6+x4*x5*x6)
eta=x7
omega*psi
psi*psiplus-(2/3)*omega^3
psi.differential()
omega*omega.differential()-psi*eta.differential()
omega^2*eta.differential()+2*psiplus*omega.differential()