Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:


*

*as a quotient of a semisimple real Lie group $G$ by a parabolic subgroup (subgroup containing a Borel),

*as an orbit of the isotropy representation (action of the point stabilizer on the tangent space) of the Riemann symmetric space associated to $G$.
Where can I find a explicit statement and proof of this equivalence, ideally in book form?  Preferably with additional details on the connection between flag manifolds and Riemann symmetric spaces (e.g., on when one is the other, and conversely)?
 A: A real flag manifold is $M=G/P$ where $G$ is a real semisimple Lie group and $P$ is a parabolic subgroup of $G$. Here parabolic means that the Lie algebra
$\mathfrak p$ contains a minimal parabolic subalgebra $\mathfrak p_{min}$ of $\mathfrak g$. 
In order to define this, let $\mathfrak g=\mathfrak k+\mathfrak s$ be a Cartan decomposition, $\mathfrak a$ is a maximal
Abelian subspace of $\mathfrak s$, 
$\mathfrak g =\mathfrak g_0+\sum_{\lambda\in\Sigma} \mathfrak g_\lambda$ be the root decomposition with respect to $\mathfrak a$.
We have $\mathfrak g_0=\mathfrak m+\mathfrak a$ where $\mathfrak m$ is the 
centralizer of $\mathfrak a$ in $\mathfrak k$. Choose a positive subsystem 
$\Sigma^+$ and let $\mathfrak n = \sum_{\lambda\in\Sigma^+}\mathfrak g_\lambda$
(nilpotent Lie algebra). We get the Iwasawa decomposition
$\mathfrak g =\mathfrak k+\mathfrak a+\mathfrak n$. Finally, 
$\mathfrak p_{min}=\mathfrak m+\mathfrak a+\mathfrak n$. 
In general, $\mathfrak p = \mathfrak p_{min} + \sum_{\lambda\in\langle\Phi\rangle}\mathfrak g_{-\lambda}$, where $\Phi$ is a subset of of the system of simple roots of $\Sigma$.  
It turns out $K$, as a subgroup of $G$, acts by left translations on $G/P$
and this action is transitive. In fact, the isotropy algebra at the basepoint
is $\mathfrak k\cap\mathfrak p$, and if you count dimensions, you see that the 
orbit is open. It is also closed by compactness of $K$, so $K/K\cap P=G/P$. 
It remains to identify the left hand side with an orbit of the isotropy representation of the symmetric space $G/K$, namely, the action of $K$ on $\mathfrak s$, but this is not hard.
I can refer you to the nice book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces", by Andreas Arvanitoyeorgos (http://www.ams.org/bookstore?fn=20&arg1=stmlseries&ikey=STML-22). 
Edit: Indeed $\mathfrak k\cap\mathfrak p =\mathfrak m + \sum_{\lambda\in\langle\Phi\rangle}(\mathfrak g_{\lambda}+\mathfrak g_{-\lambda})\cap\mathfrak k$ and you can select a point $x\in\mathfrak a\subset\mathfrak s$ 
such that $\lambda(x)=0$ if and only if $\lambda\in\langle\Phi\rangle$.
The desired $K$-orbit is that through $x$.  
