I'm studying Galois Theory on my own and already understand the correspondence between field automorphisms and subgroups of the permutation group. However, I am trying to approach it more from the historical point of view, that is in the language used by Lagrange and am having trouble finding the role of normal subgroups.
Suppose $f$ is a quartic. The strategy is to find a polynomial on the four roots $r_1,r_2,r_3,r_4$ that takes $m < deg(f)$ under the permutation of roots. These $m$ values can be obtained by solving a polynomial of lower degree and then using them recover the original roots.
What I don't understand is the following: we can use the polynomial $r_1r_2+r_3r_4$, which takes $3$ values under root permutations. A cubic is solved and then these $3$ values can be used to find $r_1,r_2,r_3,r_4$. I've seen some notes online that say that the subgroup of permutations that fix this polynomial is the Klein 4 group. But the group that fixes the polynomial is $\{I,(12),(34),(12)(34),(13)(24),(14)(23),(1423),(1324)\}$, which is not the Klein 4 group, not even a normal subgroup.
Perhaps I am mistaken to expect the normal subgroup to show up here or I misunderstood what I read online. Any help is appreciated.