TITLE: Synthetic projective lines
QUESTION [15 upvotes]: The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and any two distinct lines intersect in a unique point (plus some nondegeneracy assumptions).  There are similar notions of projective 3-space, $n$-space, and so on — but 1-dimensional projective space seems harder to capture synthetically, since there is no "room", dimensionally, for subspaces in between the points and the entire space.
Has anyone attempted to define a synthetic notion of "projective line"?  Ideally such a definition would have properties like the following:

The space $P^1(k)$ is naturally a projective line for any division ring $k$, and from any projective line $L$ satisfying enough axioms we can construct a skew field $c(L)$ such that $c(P^1(k)) \cong k$ and $P^1(c(L))\cong L$ (unnaturally).  The corresponding facts for Desarguesian projective planes are classical.
Any line in a projective plane is a projective line, and any projective line satisfying enough axioms can be embedded as a line in some projective plane.  This would be analogous to how any plane in a projective 3-space is a Desarguesian projective plane, while any Desarguesian projective plane can be embedded in a projective 3-space.

I have an idea for how one might do this, by axiomatizing the "quadrangular hexad" relation on a line in a projective plane; but before I try very hard, I'm looking for references where something like it has been tried before.

REPLY [5 votes]: You might benefit from reading Section 5.3 of John Faulkner's book The role of nonassociative algebra in projective geometry AMS 2014. The results there may have been what you had in mind by 'axiomatizing the "quadrangular hexad" relation'. These are (almost) all very old theorems, appearing in (and mostly predating) Pickert's Projektive Ebenen book from 1955, one going back to Veblen and Young from 1910 and another to von Staudt from 1860. The basic underlying idea of using quadrangular sets goes back to Desargues in 1639. (It was Desargues who introduced the term involution into mathematics, in this context, often referred to as an involution of six points.) But Desargues' work is an elaboration and extension of results of Pappus from c.340.