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

Conforming Finite Element Function Spaces in Four Dimensions, Part II: The Pentatope and Tetrahedral Prism