Question about Jacobian subalgebra

This is still is equivalent to JC.

Your equality says, $mathbb{C}[x,y]=mathbb{C}[f,g]+gmathbb{C}[x,y]$, the last term is equal to $mathbb{C}[f]+gmathbb{C}[f,g]+gmathbb{C}[x,y]=mathbb{C}[f]+gmathbb{C}[x,y]$, since $gmathbb{C}[f,g]subset gmathbb{C}[x,y]$. This says, the map $mathbb{C}[f]to mathbb{C}[x,y]/gmathbb{C}[x,y]$ is onto and then it is clear that this is an isomorphism. Then, $g=0$ is an embedded line in $mathbb{C}^2$ and by Abhyankar-Moh, is a co-ordinate line after an automorphism. Then, it is easy to verify that $mathbb{C}[f,g]=mathbb{C}[x,y]$.