A note on Clifford parallelisms in characteristic two

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.