I want to show that $(T_1T_4-T_2T_3,T_1+T_4)\subseteq k[T_1,T_2,T_3,T_4]$ is a prime ideal, where $k$ is a field.
For this, I thought of $$k[T_1,T_2,T_3,T_4]/(T_1T_4-T_2T_3,T_1+T_4)\cong k[T_1,T_2,T_3]/(T_1^2+T_2T_3)$$ via $T_4\mapsto -T_1$, and the latter is an integral domain because $T_1^2+T_2T_3\subseteq (k[T_2,T_3])[T_1]$ is prime, since it is of degree $2$ in $T_1$ and has no roots in $k[T_2,T_3]$. My question is whether this is correct, or where the mistakes are; I don't feel very confident about these arguments in general and thought a quick verification could help me. Thanks!