Are factorized parametric identities for $a A^3 + b B^3 + c C^3 = X Y Z$ known?

While studying parametric identities involving sums of three cubes, I obtained the polynomial identity in 11 variables.

The complete formulas and verification code are available here:

https://zenodo.org/records/22255821

All these identities have been checked symbolically in SymPy.

The parametric identity has the following form $$a \cdot A^3 + b \cdot B^3 + c \cdot C^3 = X \cdot Y \cdot Z$$ The identity is rather large, so I give the full expression below.

Let $S = a*d^3 + b*e^3 + c*f^3$ and $T = a*x^3 + b*y^3 + c*z^3$.

The identity takes the following form: $$a*(S*((d - h*x)*T + g*x) +g*h*(3*(b*d*e*y^2 + c*d*f*z^2 - b*e^2*x*y - c*f^2*x*z) +g*(d-h*x) -d*h*T))^3\\ + b*(S*((e - h*y)*T + g*y) +g*h*(3*(a*d*e*x^2 + c*e*f*z^2 -a*d^2*x*y- c*f^2*y*z) +g*(e - h*y) -e*h*T))^3\\ + c*(S*((f - h*z)*T + g*z) +g*h*(3*(a*d*f*x^2 + b*e*f*y^2 -a*d^2*x*z- b*e^2*y*z) +g*(f - h*z) -f*h*T))^3\\ = (a*(d - h*x)^3 + b*(e - h*y)^3 + c*(f - h*z)^3) *$$ $$(T^2*S +3*g*T*(a*d^2*x + b*e^2*y + c*f^2*z) +3*g^2*(a*d*x^2 + b*e*y^2 + c*f*z^2) + g^3) *$$ $$(T*S^2 +3*g*h*S*(a*d*x^2 + b*e*y^2 + c*f*z^2) +3*g^2*h^2*(a*d^2*x + b*e^2*y + c*f^2*z) + g^3*h^3)$$

I also obtained several related 5-variable and 6-variable parametric identities. They are not simply specializations of the 11-variable identity, but appear to form separate families.

My question is not whether the identities are correct — symbolic verification confirms them — but whether identities with this particular factorized structure are already known.

In particular, I would be interested in references concerning:

• polynomial parametrizations of equations of the form $a \cdot A^3 + b \cdot B^3 + c \cdot C^3 = X \cdot Y \cdot Z$,
• factorizations of sums of three cubes into products of similar polynomial forms;
• constructions from which identities of the above form could arise as special cases.

If this is a known construction, I would be very grateful for a reference or even a keyword/name under which it can be found. I am mainly trying to determine whether these identities are genuinely new or are a rediscovery of a classical construction.