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 $R(\Delta')$???
$\alpha$is any simple root not in$\Delta'$and it need not be orthogonal to all roots in the subsystem. $\endgroup$$\mathfrak{g} =\mathfrak{sl}_3$with$\Delta'$containing just one simple root$\beta$and$\alpha$being the other simple root. $\endgroup$