Riemannian metric of hyperbolic plane 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?
 A: I believe that this idea goes back to Minkowski, where he defines the symmetric space of SL(n) as the positive semi-definite cone (with action by similarity) - in the case of n=2, the set det = 1 is precisely the hyperbolic plane.
A: Certainly the general idea goes back at least to Minkowski, and/or Poincare, maybe Beltrami. (Poincare's model of the hyperbolic plane had geodesics arcs of circles orthogonal to the boundary, while Beltrami's geodesics were Euclidean straight lines (intersected with the disc).) 
Various of C.L. Siegel's IAS notes talk about the classical non-compact homogeneous spaces of hermitian type at great length, c. 1960, as though this were all well-known, and, indeed, his 1939 paper on what we would call "Siegel modular forms" seems to take the general idea for granted.
Jacques Tits has several papers (look at MathSciNet) that use the octonions to model the exceptional domains and groups. I remember first seeing such examples in the book of Walter Bailey, Jr. "Introductory Lectures on Automorphic Forms".
See also J.P. May's little book on "Exceptional Lie groups/algebras"...
The Wiki entry on "classical groups" gives a good outline of many things... One point is that, although E. ("Poppa") Cartan and Killing had models of the "classical" Lie groups and related things c. 1890, the exceptional groups and domains did not have known models until perhaps Chevalley in the late 1940s.
In courses of G. Shimura at Princeton in the mid-1970s, the classical groups and domains were mentioned as common-places... without references, mostly.
