I am looking for a proof of the fact that the ideal $$ I = (x^2 + y^2 + z^2 + w^2, ax^2 + by^2 + cz^2 + dw^2) $$ with $a, b, c, d$ different numbers, is prime in $\mathbb{C} [x, y, z, w] $.
I know that a homogeneous ideal is prime if and only if for every couple of homogeneous polynomials $f, g$, happens that $fg \in I \Rightarrow f\in I $ or $g \in I$, but this doesn't help. I tried to verify that the associated variety is irreducible but I didn't succeed. Any idea? Thank u.