Let us define the real hyperbolic plane $\mathbb{RH}^2$ to be the dual symmetric space of non-compact type for the compact symmetric space $S^2=SO(3)/SO(2)$. For a symmetric space $G/K$ with Cartan decomposition $\mathfrak{g=k \oplus p}$, the dual symmetric space is given as the space $G'/K$ which has Cartan decomposition $\mathfrak{g'=k \oplus \sqrt{-1} p}$. In the case of $S^2$, i.e. $G=SO(3)$ and $K=SO(2)$, this leads to $G' \simeq SO(2,1)$ and the symmetric space $SO(2,1)/SO(2)$. So, to summarize: $\mathbb{RH}^2=SO(2,1)/SO(2)$. (This argument is presented, for example, in Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, p. 238)
Now here is what confuses me: I have seen $\mathbb{RH}^2$ as a homogenous space denoted as $SL(2)/SO(2)$, for example here: Berndt, Console, Enrique: Submanifolds and Holonomy, Second Edition, p. 350. $SL(2)$ and $SO(2,1)$ are not isomorphic, despite the exceptional isomorphism $\mathfrak{so}(3) \simeq \mathfrak{sl}(2)$. So, my question is:
What is $\mathbb{RH}^2$ as a homogenous space?
It is not true that $H^2= SO(2,1)/SO(2)$, as $SO(2,1)$ is not connected. Furthermore the action of $SL(2,R)$ on $H$ has a kernel, $\pm Id$, so that $H^2= PSL(2,R)/PSO(2)$, not $SL(2)/SO(2)$
In fact $PSL(2,R)$ is isomorphic to $SO_0(2,1)$ by the following argument : $SL(2,R)$ acts on the vector of $(3,3)$ symmetric real matrices. This vector space has a natural invariant quadratic form of signature $(2,1)$, and the kernel of the action is $\pm Id$. In this langage the hyperbolic plane is identify with the set of positive definite quadratic form with discrimant 1.
One checks that the map $PSL(2,R)$ to $S0 _0(2,1)$ is a bijection using the following observation. A positive definite quadratic form can be written as $q(x,y) = x^2+ y^2$ in some base of determinant 1. In this base, the Cartan involution around $q$ is the image in PSL(2) of the linear map $x\to y$, $y\to -x$.
The image of $PSL(2)$ contains all Cartan involutions, and is therefore surjective ; injectivity is easy to check.