Assume that $\mathbb{F}$ is not of characteristic $2$ or $3$ and consider the dehomogenization $X^3+Y^3+1=0$. Use the substitutions $X=\frac{2−v}u$ and $Y=\frac{1+v}u$ to get an equation in Weierstrass form. Then transform the equation to reduced Weierstrass form.
These substitutions lead to $9(v^2-v+1)+u^3=0$. I don't know how to get rid of the $v$ term, so that I would get an equation of the form $v^2=u^3+Au+B$. I thought about solving for $v$ and then square it, but I get $v_{1,2}=\frac{3\pm\sqrt{-27-4u^3}}6$, thus $v_{1,2}^2=-\frac23u^3\pm\sqrt{-27-4u^3}-3$, where the second term is not of degree $1$ but of degree $\frac32$.
What should I do?