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Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying

$$\Delta^+ = - \Delta^-\tag{$*$}\label{475103_star}$$

and such that for any two roots $\alpha,\beta \in \Delta^+$ such that $\alpha + \beta \in \Delta$ it holds that

$\alpha + \beta \in \Delta^+$.

Is there a name for a decomposition into subsets $\Delta^+$ and $\Delta^-$ that only satisfy \eqref{475103_star}?

Do such decompositions arise naturally in the study of, or applications of, root systems?

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    $\begingroup$ I personally have used such a decomposition \eqref{475103_star} to construct a distinguished map from $\mu_2$ to the centre of a reductive group, defined as $\sum_{\alpha \in \Delta^+} \alpha^\vee$ (I think that I learned this construction from Kottwitz); but the reason that I could use them was that the choice made no difference—it might as well have been a set of positive roots. Since these sets make almost no use of the structure of a root system, it would seem surprising to me to see them used extensively in root-systems-specific contexts. Where did you encounter them? $\endgroup$
    – LSpice
    Commented Jul 15 at 18:44

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