#357B It does not have (purely) coclosed G2-structure. Second obstruction. Corollary 5.5. Compute closed 4-forms

A.<x1,x2,x3,x4,x5,x6,x7> = GradedCommutativeAlgebra(QQ)
M=A.cdg_algebra({x3: x1*x2,x5: x1*x3, x6:x2*x3, x7:x1*x4})
M.inject_variables()
M.cohomology(4)

(x1*x2*x3).differential()
(x1*x2*x4).differential()
(x1*x2*x5).differential()
(x1*x2*x6).differential()
(x1*x2*x7).differential()
(x1*x3*x4).differential()
(x1*x3*x5).differential()
(x1*x3*x6).differential()
(x1*x3*x7).differential()
(x1*x4*x5).differential()
(x1*x4*x6).differential()
(x1*x4*x7).differential()
(x1*x5*x6).differential()
(x1*x5*x7).differential()
(x1*x6*x7).differential()
(x2*x3*x4).differential()
(x2*x3*x5).differential()
(x2*x3*x6).differential()
(x2*x3*x7).differential()
(x2*x4*x5).differential()
(x2*x4*x6).differential()
(x2*x4*x7).differential()
(x2*x5*x6).differential()
(x2*x5*x7).differential()
(x2*x6*x7).differential()
(x3*x4*x5).differential()
(x3*x4*x6).differential()
(x3*x4*x7).differential()
(x3*x5*x6).differential()
(x3*x5*x7).differential()
(x3*x6*x7).differential()
(x4*x5*x6).differential()
(x4*x5*x7).differential()
(x4*x6*x7).differential()
(x5*x6*x7).differential()

Defining x1, x2, x3, x4, x5, x6, x7
Free module generated by {[x1*x3*x4*x5], [x2*x3*x4*x5], [x2*x3*x4*x6], [x1*x3*x5*x6], [x2*x3*x5*x6], [x1*x4*x5*x6 - x2*x3*x5*x7], [x2*x4*x5*x6 - x2*x3*x6*x7], [x1*x3*x5*x7], [x1*x4*x5*x7], [x2*x4*x5*x7 + x1*x4*x6*x7], [x2*x4*x6*x7]} over Rational Field
0
0
0
0
0
0
0
0
0
0
x1*x2*x3*x4
0
x1*x2*x3*x5
0
-x1*x2*x3*x7
0
0
0
x1*x2*x3*x4
-x1*x2*x3*x4
0
0
x1*x2*x3*x6
-x1*x2*x4*x5 + x1*x2*x3*x7
-x1*x2*x4*x6
x1*x2*x4*x5
x1*x2*x4*x6
x1*x2*x4*x7
x1*x2*x5*x6
-x1*x3*x4*x5 + x1*x2*x5*x7
-x1*x3*x4*x6 + x1*x2*x6*x7
x2*x3*x4*x5 - x1*x3*x4*x6
-x1*x3*x4*x7
-x2*x3*x4*x7
x1*x4*x5*x6 - x2*x3*x5*x7 + x1*x3*x6*x7