Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27

Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a 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 ideals $J=\operatorname{gin}_{\operatorname{lex}}(I)$ are $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ are $(x_2^3,x_1x_2,x_1^2)$.

How to find it?

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27

Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a 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 ideals $J=\operatorname{gin}_{\operatorname{lex}}(I)$ are $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ are $(x_2^3,x_1x_2,x_1^2)$.

How to find it?