  In this paper, we present explicit expressions for conforming finite element
function spaces, basis functions, and degrees of freedom on the pentatope and
tetrahedral prism elements. More generally, our objective is to construct
finite element function spaces that maintain conformity with
infinite-dimensional spaces of a carefully chosen de Rham complex. This paper
is a natural extension of the companion paper entitled "Conforming Finite
Element Function Spaces in Four Dimensions, Part I: Foundational Principles and
the Tesseract" by Nigam and Williams, (2023). In contrast to Part I, in this
paper we focus on two of the most popular elements which do not possess a full
tensor-product structure in all four coordinate directions. We note that these
elements appear frequently in existing space-time finite element methods. In
order to build our finite element spaces, we utilize powerful techniques from
the recently developed 'Finite Element Exterior Calculus'. Subsequently, we
translate our results into the well-known language of linear algebra (vectors
and matrices) in order to facilitate implementation by scientists and
engineers.
