Clifford Parallelisms and External Planes to the Klein quadric

Abstract

For any three-dimensional projective space ${\mathbb P}(V)$, where $V$ is a vector space over a field $F$ of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of ${\mathbb P}(V)$ and those planes of ${\mathbb P}(V\wedge V)$ that are external to the Klein quadric representing the lines of ${\mathbb P}(V)$. We also give two characterisations of a Clifford parallelism of ${\mathbb P}(V)$, both of which avoid the ambient space of the Klein quadric.