number theory - Lamé's proof of Fermat Last Theorem for n=3

Among the great mathematicians who struggled to try and figure out a proof of Fermat's last theorem in 19th century, Lamé was probably one of the most convinced in having succeded. I first came across this topic when reading the book $\textit{Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory}$. After a brief search for details of Lamé's proof, I found in What was Lame's proof? the steps he followed, but, even in its fault, it does not work if $n=3$, as the author himself claimed, in $x^n+y^n=z^n$, the equation in Fermat's Last Theorem statement.

I found that the result for $n=3$ was already proven to be true at the time, thanks only to the factorization of the left member of the equation, so Lamé did not actually needed to prove it. However, in many other sources, it is stated that there is a different proof which exploits the infinite descent method, like the one Lamé employed, but no one gives details about that.

The closest I came was in some notes related to a course of algebraic number theory. They stated that, working in $\mathbb{Z}[j]$, where $ j=\frac{-1+i\sqrt{3}}{3} $ is a 3rd-root of the unity in the complex field, $x^3 + y^3 = z^3$ can be rewritten as follows: $$ \left( \frac{x}{z} \right)^3 -1 = - \left( \frac{y}{z} \right)^3. \tag{1} \label{1} $$ So, by using $$X^3-1 = (X - 1)(X^2 + X + 1) = (X - 1)(X - j)(X - j^2),$$ and multiplyng by $z^3$ in $(\ref{1})$, it ensues that: $$ (x - z)(x - jz)(x -j^2z) = -y^3. $$

Unfortunately, the notes ended by claiming that, from this identity, Lamé concluded by infinite descent using the additional fact that, if the product of three coprime integers is a cube, then each of the factor can be written as a cube. Here, the three coprime elements should be $ (x - z)$, $(x - jz)$ and $(x -j^2z)$. Probably, Lamé made a mistake in extending to $\mathbb{Z}[j]$ such a property known to be true only in the integer ring, still I can't understand how the infinite descent should develop, starting from this point. Can anyone please help me to find a way in proceeding with this argument?