Changing Parent on multivariable polynomial ring

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This question is similar to https://ask.sagemath.org/question/803...

R.<x,y,z,w> = QQ[]
f=x*y*z*w
f1=derivative(f,x)
f2=derivative(f,y)
f3=derivative(f,z)
f4=derivative(f,w)
J = R.ideal([f1, f2, f3,f4])

Now f is in its Jacobian by Euler identity so we can lift f as follows.

f in J

returns true.

f.lift(J)

returns the lift of f. Now consider h=f^(2)/ (xyz). Although this seems like a rational function, this is actually a polynomial.

h in J

returns true. However, I get an error when doing

h.lift(J)

The reason for this is Sage reads h as living not in the multivariable ring, but the fraction field of it because I divided by xyz. Apparently, there is no lift function for polynomials living in the fraction field. However, it is still an honest polynomial as it divides cleanly with no remainder. To be more clear,

f.parent()

returns Multivariate Polynomial Ring in x, y, z, w over Rational Field.

h.parent()

returns Fraction Field of Multivariate Polynomial Ring in x, y, z, w over Rational Field. Is there a way to make h belong to the multivariate ring instead of its fraction field so I can apply the lift function?