It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space $P^7$. This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discuss the extended inverse kinematic map, motions corresponding to straight lines in $P^7$ and linear complexes of SE(3)-displacements. Moreover we present an application of this interpretation in the context of interactive motion design.