  We present a systematic approach to finding Higgs vacuum expectation values,
which break a symmetry G to differently embedded isomorphic copies of a
subgroup $H \subset G$ . We give an explicit formula for recovering each point
in the vacuum manifold of a Higgs field which breaks G -> H. In particular we
systematically identify the vacuum manifold G/H with linear combinations of the
vacuum expectation values breaking G -> H_1 -> ... -> H_l. We focus on the most
applicable case for current work on grand unified theories in extra dimensional
models and low-energy effective theories for quantum chromodynamics. Here the
subgroup, H, stabilizes the highest weight of the fundamental representation
leading to a simple expression for each element of the vacuum manifold;
especially for an adjoint Higgs field. These results are illustrated explicitly
for adjoint Higgs fields and clearly linked to the mathematical formalism of
Weyl groups. We use the final section to explicitly demonstrate how our work
contributes to two contemporary high-energy physics research areas.
