I am looking for a reference for the following admittedly imprecise statement:



> 
> Any projective invariant of n points in the projective plane may be
> expressed as a function of well-chosen cross-ratios.
> 
> 
> 


By *projective invariant* I mean a rational function defined on the set of $n$-tuple of distinct points in the projective plane on an arbitrary field $K$, invariant under the action of the projective group of transformations.
This is a folklore result that is often stated without proof (e.g. in Efimov, *higher geometry*) but I can't find a reference providing a precise statement and a proof.



Projective invariants of the plane and cross ratio
I believe the precise statement you are looking for is proposition 3.3 on page 26 of Danylo Radchenko's doctoral dissertation, [*Higher cross-ratios and geometric functional equations for polylogarithms*](https://core.ac.uk/download/pdf/223015944.pdf) (Rheinischen Friedrich-Wilhelms-Universität Bonn, 2016): the proof of the proposition explains how to view the variables of the invariant field as projected cross-ratios of the points.


