  It is well known that a purely inseparable field extension $L/F$ with some
extra property and degree $[L:F]=4$ determines a Clifford parallelism on the
set of lines of the three-dimensional projective space over $F$. By extending
the ground field of this space from $F$ to $L$, we establish the following
geometric description of such a parallelism in terms of a distinguished
`absolute pencil of lines' of the extended space: Two lines are Clifford
parallel if, and only if, there exists a line of the absolute pencil that meets
both of them.
