14-Piece Dissection of Juel's Pyramid into a Parallelepiped

The base of Juel's pyramid is a cube and its apex is the center of the cube. A dissection of the pyramid into a parallelepiped is described in [2]. The construction consists of cutting the pyramid into three layers.
There is a hinged dissection of the Hill tetrahedron into a triangular prism, using a construction found independently by P. Schobi and A. Hanegraaf [1]. This is a three-piece dissection. Together these yield a four-piece dissection of Juel's pyramid into the prism.
[1] G. N. Frederickson, Dissections: Plane & Fancy, New York: Cambridge University Press, 2002 p. 235.
[2] H. Meschkowski, Grundlagen der Euklidischen Geometrie (Croatian edition), Zagreb: Skolska Knjiga, 1978 pp. 211–214.