  We discuss two approaches for the formulation and implementation of
space-time discontinuous Galerkin spectral element methods (DG-SEM). In one,
time is treated as an additional coordinate direction and a Galerkin procedure
is applied to the entire problem. In the other, the method of lines is used
with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in
time. The two approaches are mathematically equivalent in the sense that they
lead to the same discrete solution. However, in practice they differ in several
important respects, including the terminology used to describe them, the
structure of the resulting software, and the interaction with nonlinear
solvers. Challenges and merits of the two approaches are discussed with the
goal of providing the practitioner with sufficient consideration to choose
which path to follow. Additionally, implementations of the two methods are
provided as a starting point for further development. Numerical experiments
validate the theoretical accuracy of these codes and demonstrate their utility,
even for 4D problems.
