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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.

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