  We study the geometric engineering of gauge theories with gauge group Spin(4)
and SO(4) using crepant resolutions of Weierstrass models. The corresponding
elliptic fibrations realize a collision of singularities corresponding to two
fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight
different ways to engineer such collisions using decorated Kodaira fibers. The
Mordell-Weil group of the elliptic fibration is required to be trivial for
Spin(4) and Z/2Z for SO(4).
  Each of these models have two possible crepant resolutions connected by a
flop. We also compute a generating function for the Euler characteristic of
such elliptic fibrations over a base of arbitrary dimensions. In the case of a
threefold, we also compute the triple intersection numbers of the fibral
divisors. In the case of Calabi-Yau threefolds, we also compute their Hodge
numbers, and check the cancellations of anomalies in a six-dimensional
supergravity theory.
