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

$~~~ \Delta^+ = - \Delta^- ~~~~~~~~(*)$

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 (*)?

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

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?