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I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible symmetric spaces. In page 9, proposition 2.2, there is a criterion for $\psi\in \mathfrak r_{\psi}$, but is it not always the case given the way $\mathfrak r_\psi$ is defined in page 8?

Maybe I am missing something. Can someone please clarify? 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.

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  • $\begingroup$ It would probably be a good idea to expand your question to say what the criterion is, what $\psi$ is, and the definition of $\mathfrak{r}_\psi$ so that a reader doesn't have to click through to the link to understand what you are asking. $\endgroup$
    – Callum
    Commented Feb 1, 2023 at 9:00
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    $\begingroup$ From a quick glance, $\psi$ is a restricted root and $\mathfrak{r}_\psi$ is a subset of the (not restricted) root system. So $\psi$ does not have to be an actual root (if it is then it must be in $\mathfrak{r}_\psi$) and the condition is precisely asking when $\psi$ is a root. $\endgroup$
    – Callum
    Commented Feb 1, 2023 at 9:10

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