A doubt from Araki's 1962 paper on classification of irreducible symmetric spaces I am looking at Soho Araki's 1962 paper for the classification of real semisimple lie algebras.Here's the link to the paper: Araki's paper.In  page $9$, proposition $2.2$,there is a criteria for $\psi\in r_{\psi}$, but is it not always the case given the way $r_\psi$ is defined in page $8$?
Maybe I am missing something. Can someone please clarify? Thanks in advance.
Also, can someone refer me to something where the methods discussed in this paper are illustrated with concrete examples? That would be of great help to me.
 A: A concrete example will help: Take as our real Lie algebra
$\mathfrak{g} = \mathfrak{su}_{1,2} := \lbrace
\begin{pmatrix}
 a+bi & c+di & ei\\
 f+gi & -2bi & -c+di\\
 hi & -f+gi & -a+bi
\end{pmatrix} : a, ..., h \in \mathbb{R} \rbrace$.
Its complexification $\mathfrak g_{\mathbb C}$ is isomorphic to $\mathfrak{sl}_3(\mathbb C)$ i.e. those $3\times 3$ complex matrices with trace $0$. In Araki's terminology, we can choose as $\mathfrak h_{\mathbb C}$ the diagonal matrices in $\mathfrak{sl}_3(\mathbb C)$. If
$\alpha_1 : diag(x_1,x_2,x_3) \mapsto x_1-x_2$ and $\alpha_2 : diag(x_1,x_2,x_3) \mapsto x_2-x_3$
then the root system for this $\mathfrak h_{\mathbb C}$ contains exactly the roots in $\mathfrak r = \{\pm \alpha_1, \pm \alpha_2, \pm (\alpha_1 +\alpha_2)\}$.
Now, Araki's $\mathfrak h_0$ (defined in 2.3) is just the traceless diagonal matrices with all real entries. Note that this is not a subset of the $\mathfrak g$ we started with. In fact,
$$\mathfrak h_0 \cap \mathfrak g =  \{ \begin{pmatrix}
 a & 0 & 0\\
 0 & 0 & 0\\
 0 & 0 & -a
\end{pmatrix} : a \in \mathbb{R} \rbrace = \mathfrak h_0^-$$
while
$$\mathfrak h_0^+ = \{ \begin{pmatrix}
 c & 0 & 0\\
 0 & -2c & 0\\
 0 & 0 & c
\end{pmatrix} : c \in \mathbb{R} \rbrace .$$
(You can check for yourself that complex conjugation on $\mathfrak g_{\mathbb C}$, under the iso $\mathfrak g_C \simeq \mathfrak{sl}_3(\mathbb C)$, operates via
$$\sigma (\begin{pmatrix}
 x_1 & x_2 & x_3\\
 x_4 & x_5 & x_6\\
 x_7 & x_8 & x_9
\end{pmatrix}) = \begin{pmatrix}
 -\bar x_9 & -\bar x_6 & -\bar x_3\\
 -\bar x_8 & -\bar x_5 & -\bar x_8\\
 -\bar x_7 & -\bar x_6 & -\bar x_1
\end{pmatrix}.)$$
Now look at page 6 of the paper. In the proof of prop 2.1, we "identify $(\mathfrak h_0^-)^\ast$ with a subspace of $\mathfrak h_0^\ast$ which is the annihilator of $\mathfrak h_0^+$."
Now $\mathfrak h_0^\ast$ is just the real span of the roots, and in our example, the annihilator of $\mathfrak h_0^+$ is the one-dimensional subspace in there spanned by $\alpha_1 +\alpha_2$. That is, with the notation of the paper,
$\psi \in \mathfrak r_\psi$ if and only if $\psi$ annihilates $\mathfrak h_0^+$.
In our case, this is true for $\psi = \pm (\alpha_1 + \alpha_2)$, but not for any other root. In fact, using Araki's notation on our example, we have
$$\mathfrak r_0 = \{\alpha \in \mathfrak r: \alpha_{\vert \mathfrak h_0^-} = 0\} =  \emptyset$$
and if we call $\beta := \alpha_1 + \alpha_2$ then $\mathfrak r^-$, the restrictions of all roots to $\mathfrak h_0^-$, consists of
$$\mathfrak r^- = \{\pm \beta, \pm \frac12 \beta \},$$
the latter being the respective restrictions of both $\pm \alpha_1$ and $\pm \alpha_2$.
In fact, $\sigma(\alpha_1) = \alpha_2$, and you can see now that as said above, $\beta \in r_\beta$ and we are in case a of bottom of page 6. But for $\psi=\frac12 \beta$ we have
$$\mathfrak r_{\frac12 \beta} = \{\alpha_1, \alpha_2\}$$
and here we are in case c at the top of page 7. (Case b does not occur in this example.)
The propositions on page 9 should make sense now too.

I treated very similar things in my thesis, in particular this is basically example 3.2.9 in there. Araki's $r^-$ is equivalent to the standard definition of "$k$-rational root systems" (cf. https://math.stackexchange.com/a/3349137/96384 and links from there; in the example above, we see $\mathfrak r^-$ is of type $BC_1$) although I, following others (Tits and Satake in particular), would define it slightly differently: I do not like that identification of a quotient with a subspace too much. Quotients of root systems are not yet en vogue, but under the name "foldings" they are gaining popularity, or so I like to believe.
