Noether normalization of the quadratic surface xy plus yz plus zx

Letover a field in which $3$ is invertible. Put $u=x-z$ and $v=y-z$. ThensoDivision by this monic quadratic makes $T$ a free $k[u,v]$-module with basis $1,z$. Thus $k[u,v]$ is a polynomial algebra and $T$ is integral over it.