  We propose a construction of non-trivial vacua for Yang-Mills theories on the
3-torus. Although we consider theories with periodic boundary conditions,
twisted boundary conditions play an essential auxiliary role in our
construction. In this article we will limit ourselves to the simplest case,
based on twist in SU(2) subgroups. These reproduce the recently constructed new
vacua for SO(N) and G_2 theories on the 3-torus. We show how to embed the
results in the other exceptional groups F_4 and E_{6,7,8} and how to compute
the relevant unbroken subgroups. In a subsequent article we will generalise to
SU(N > 2) subgroups. The number of vacua found this way exactly matches the
number predicted by the calculation of the Witten index in the infinite volume.
