We describe a new algorithm, based on sieving procedures, for determining the minimal index and all elements with minimal index in a class of totally real quartic fields with Galois group $D_8$. It is not universally applicable, but its applicability is easily checked for any particular example, and it is very fast when applicable. We include several tables demonstrating the potential of the method. (A more general approach for quartic fields, described in [Gaál et al.], requires much more computation time for each field.)
Finally, we present a family of totally real quartic fields with Galois group $D_8$ and having minimal index 1 (that is, a power integral basis).
"On the resolution of index form equations in dihedral quartic number fields." Experiment. Math. 3 (3) 245 - 254, 1994.