#n10. 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=e3*e5*e7+e1*e2*e5+e1*e3*e4-e2*e4*e7
psif=f3*f5*f7+f1*f2*f5+f1*f3*f4-f2*f4*f7
psig=g3*g5*g7+g1*g2*g5+g1*g3*g4-g2*g4*g7
psih=h3*h5*h7+h1*h2*h5+h1*h3*h4-h2*h4*h7
	
(-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=e3*e5*e7+e1*e2*e5+e1*e3*e4-e2*e4*e7
psif=f3*f5*f7+f1*f2*f5+f1*f3*f4-f2*f4*f7
psig=g3*g5*g7+g1*g2*g5+g1*g3*g4-g2*g4*g7
psix=x3*x5*x7+x1*x2*x5+x1*x3*x4-x2*x4*x7
psiy=y3*y5*y7+y1*y2*y5+y1*y3*y4-y2*y4*y7
psiz=z3*z5*z7+z1*z2*z5+z1*z3*z4-z2*z4*z7
psixyz=x3*y5*z7+x1*y2*z5+x1*y3*z4-x2*y4*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*x4-x1*x3*x5+x3*x4*x7+x2*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=e3*e5*e7+e1*e2*e5+e1*e3*e4-e2*e4*e7
psif=f3*f5*f7+f1*f2*f5+f1*f3*f4-f2*f4*f7
omegaxf=(x1*f7+4/5*(x2*f7-x1*f3)-2*x2*f3-x4*f5+3/5*(x4*f7-x1*f5))-(f1*x7+4/5*(f2*x7-f1*x3)-2*f2*x3-f4*x5+3/5*(f4*x7-f1*x5))
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 + 8/5x1*x2 + 6/5x1*x4 + 2x2^2 + 2x3^2 + 8/5x3*x7 + x4^2 + x5^2 - 6/5x5*x7 + x7^2
# [1,4/5,0,3/5,0,0],[4/5,2,0,0,0,0],[0,0,2,0,0,-4/5],[3/5,0,0,1,0,0],[0,0,0,0,1,-3/5],[0,0,-4/5,0,-3/5,1]
# 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:x2*x3 })
M.inject_variables()
omega=x1*x7+4/5*(x2*x7-x1*x3)-2*x2*x3-x4*x5+3/5*(x4*x7-x1*x5)
psi=x3*x5*x7+x1*x2*x5+x1*x3*x4-x2*x4*x7
psiplus=16/25*(x1*x2*x4-x1*x3*x5+x3*x4*x7+x2*x5*x7)
eta=x6+4/5*x5
omega*psi
psi*psiplus-(2/3)*omega^3
psi.differential()
omega*omega.differential()-psi*eta.differential()
omega^2*eta.differential()+2*psiplus*omega.differential()