*I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space.*

One can get a formula for the Fubini-Study metric on the complex projective plane via a pullback of the scalar product $\langle A , B \rangle = \mathrm{Tr}\, AB$ on the space of hermitian matrices along the mapping $x \mapsto \frac{x x^\dagger} {x^\dagger x}$.

I have extended this calculation to the case of the octonionic projective plane plane and in the process I've noticed that it actually works also for the hyperbolic plane and indefinite signatures in general. One just has to use $G$-hermitian matrices, for some symmetric bilinear form $G$. Explicitely, one can obtain the hyperbolic metric via pullback along $$ x \mapsto \frac{x(Gx)^\dagger}{(Gx)^\dagger x}. $$ The target space is the space of matrices that satisfy $A^\dagger G = GA$ and the scalar product $\langle A , B \rangle = \mathrm{Tr}\, AB$ is no longer necessarily positive definite.

I suspect this is well known in the classical case but it's very hard to google references because there is so much material. Moreover, I suspect this first appeared in the 19th century.

**Q1:** What should I cite as a reference for the classical case of (real or complex) hyperbolic plane?

**Q2:** Has this construction been actually used in the octonionic (or quaternionic) setting before?