  The stability, robustness, accuracy, and efficiency of space-time finite
element methods crucially depend on the choice of approximation spaces for test
and trial functions. This is especially true for high-order, mixed finite
element methods which often must satisfy an inf-sup condition in order to
ensure stability. With this in mind, the primary objective of this paper and a
companion paper is to provide a wide range of explicitly stated, conforming,
finite element spaces in four-dimensions. In this paper, we construct explicit
high-order conforming finite elements on 4-cubes (tesseracts); our construction
uses tools from the recently developed `Finite Element Exterior Calculus'. With
a focus on practical implementation, we provide details including Piola-type
transformations, and explicit expressions for the volumetric, facet, face,
edge, and vertex degrees of freedom. In addition, we establish important
theoretical properties, such as the exactness of the finite element sequences,
and the unisolvence of the degrees of freedom.
