  We find new supersymmetric four-dimensional Minkowski flux vacua of type II
string theory on nilmanifolds and solvmanifolds. We extend the results of M.
Grana, R. Minasian, M. Petrini, and A. Tomasiello to the case of intermediate
SU(2) structures (the two internal supersymmetry parameters are neither
parallel nor orthogonal). As pointed out recently by P. Koerber and D. Tsimpis,
intermediate SU(2) structures are possible when one considers "mixed"
orientifold projection conditions. To find our vacua, we rewrite these
projection conditions in a more tractable way by introducing new variables: the
projection basis. In these variables, the SUSY conditions become also much
simpler to solve, and we find three new vacua. In addition, we find that these
variables correspond to the SU(2) structure appearing with the dielectric pure
spinors, objects introduced and discussed by R. Minasian, M. Petrini, A.
Zaffaroni, and N. Halmagyi, A. Tomasiello, in the AdS/CFT context. Besides, our
solutions provide some intuition on what a dynamical SU(3) x SU(3) structure
solution could look like.
