  We study the geometry of elliptic fibrations given by Weierstrass models
resulting from Step 6 of Tate's algorithm. Such elliptic fibrations have a
discriminant locus containing an irreducible component $S$, over which the
generic fiber is of Kodaira type I$^*_0$. In string geometry, these geometries
are used to geometrically engineer G$_2$, Spin($7$), and Spin($8$) gauge
theories. We give sufficient conditions for the existence of crepant
resolutions. When they exist, we give a complete description of all crepant
resolutions and show explicitly how the network of flops matches the Coulomb
branch of the associated gauge theories. We also compute the triple
intersection numbers in each chamber. Physically, they correspond to the
Chern-Simons levels of the gauge theory and depend on the choice of a Coulomb
branch. We determine the representations associated with these elliptic
fibrations by computing intersection numbers with fibral divisors and then
interpreting them as weights of a representation. For a five-dimensional gauge
theory, we compute the number of hypermultiplets in each representation by
matching the triple intersection numbers with the superpotential of the theory.
  We also discuss anomaly cancellations of a six-dimensional supergravity
theory obtained by a compactification of F-theory on an elliptically fibered
Calabi--Yau threefold corresponding to a G$_2$, Spin($7$), or Spin($8$) gauge
theory.
