Took me a while to understand why this works (e.g. $z$ exists and has the desired properties). I have some questions left: 1) How do you know that you picked the right $z$ without knowing the connected components? 2) You do this calculation with a fixed $N$ and increase it step by step until you found $z$, right? If yes: Can you bound $N$ (maybe by the degree of $f_i$)?

I didn't get yet, how one can use the levi decomposition to see (the non-trivial part of) $G^u=$intersection of kernels$=:H$. Aside from that, that was the answer i was looking for, thanks. Let me fill in some more details, which were helpful to me: There are finitely many of those $g_j$ , since $G/H$ is diagonalizable. (Assuming the representation from above there are n those characters.) Using the isomorphism $X(G)=X(G/H)$ , one should see that the relations necessary are of degree $1$.