TITLE: Sums of two cubes
QUESTION [13 upvotes]: In his solution of the equation $x^3 + dy^3 = 1$, Nagell
comes across the equation
$$ u^3 + 6u^2v + 3uv^2 - v^3 = w^3. $$
He then observes that 
$$ (u^3 + 6u^2v + 3uv^2 - v^3) U^3 = V^3 + W^3 $$
for
$$ U = u^2+uv+v^2, \quad
   V = u^3+3u^2v-v^3, \quad
   W = 3u^2v+3uv^2, $$
and then appeals to Fermat's Last Theorem for the exponent $3$.
My question: what is known about representing a binary cubic form
$$ au^3 + bu^2v + cuv^2 + dv^3 $$
as a sum of two rational cubes in the polynomial ring ${\mathbb Z}[u,v]$?
Where is Nagell's solution coming from?
P.S. Dehomogenizing by setting $v = 1$ gives us an elliptic surface. Still I wonder how Nagell dreamed up his solution.

REPLY [3 votes]: A paper by Bruce Reznick and Jeremy Rouse, preprint available at http://front.math.ucdavis.edu/1012.5801 completely solves the equation $x^3+y^3=f^3+g^3$, where
$f$ and $g$ are homogeneous rational functions in $x$ and $y$.