Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra, pp. 26-27.
Let $f,g\in k[x_1,x_2,x_3,x_4]$ be generic forms of degree $d$ and $e$. The generic initial ideal of $I=\langle f,g\rangle$ for both the lexicographic order and the inverse lexicographic order when $(d,e)=(2,2)$: the ideal $J=\operatorname{gin}_{\operatorname{lex}}(I)$ is $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ is $(x_2^3,x_1x_2,x_1^2)$.
How to find it?