Role of normality of the Klein group in solving a quartic using Lagrange resolvents

Question

There are several ways how a useful resolvent for solving a quartic equation can be defined, e.g. (the roots are a, b, c, d)

1. (a + b)(c + d)
2. ab + cd
3. Let X = a + bi - c - di, then we take X * complex_conjugate(X)

All these expressions have an orbit of size 3 under S4 and actually the same orbit under the Klein group V4. Since V4 is the normal subgroup that "saves" the solvability of A4 (therefore S4), I wonder whether there's an explanation for the role its normality plays in existence of the resolvents given above.