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Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, \Delta'\subset \Delta and R(\Delta')=R\cap \mathbb Z(\Delta'). Define p(\Delta')=\mathfrak h \bigoplus_{\alpha \in R(\Delta')} \mathfrak g_{\alpha} \bigoplus_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g_{\alpha} the parabolic subalgebra associated to \Delta'.

If $\alpha$ is a simple root in R^+(\Delta)\setminus R^+ (\Delta'), then $\beta(h_\alpha)=0$ for all $\beta$ in R(\Delta')???

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The answer is NO. Take $g=sl(3)$ and $\Delta=\{a_1,a_2\}$. The unique choice for \Delta' is $\{a_2\}$. The result is clearly false in this scenery.