Is there a theorem which characterises which bijective maps $f:\mathbb{GP}^1 \to \mathbb{GP}^1$, where $\mathbb{G:F}$ is a field extension of degree 2, have the property that whenever the cross ratio of four points of $\mathbb{GP}^1$, $(a,b;c,d)$, is in $\mathbb{FP}^1$, then the same is true for $(f(a),f(b);f(c),f(d))$?
For instance when $\mathbb G$ is the complex numbers and $\mathbb F$ is the real numbers, then the set of such $f$ forms the set of Moebius and anti-Moebius transformations. This seems to be related to the Fundamental Theorem of Projective Geometry. But it's for a 1-dimensional projective space over $\mathbb G$. Unfortunately, the Fundamental Theorem needs at least 2 dimensions.
A route to an answer: It's possible to embed $\mathbb {GP^1}$ in $\mathbb {FP^3}$ as some quadric $Q$. Call the embedding map $i:\mathbb{GP^1} \to \mathbb{FP}^3$. The property that the cross ratio of $(a,b;c,d)$ (for $a, b, c, d \in \mathbb{GP}^1$) is in $\mathbb{FP}^3$ should translate to the claim that the set $\{i(a),i(b),i(c),i(d)\}$ lies on the intersection of $Q$ and some projective plane. This in turn implies that the projective planes intersecting $Q$ are permuted by $f$. But there are projective planes not intersecting $Q$. I think pole/polar type stuff can finish.