  Let $F$ be Cayley's ruled cubic surface in a projective three-space over any
commutative field $K$. We determine all collineations fixing $F$, as a set, and
all cubic forms defining $F$. For both problems the cases $|K|=2,3$ turn out to
be exceptional. On the other hand, if $|K|\geq 4$ then the set of simple points
of $F$ can be endowed with a non-symmetric distance function. We describe the
corresponding circles, and we establish that each isometry extends to a unique
projective collineation of the ambient space.
