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. – Jim Humphreys Dec 4 '10 at 20:55`$\mathfrak{g} =\mathfrak{sl}_3$`

with`$\Delta'$`

containing just one simple root`$\beta$`

and`$\alpha$`

being the other simple root. – Jim Humphreys Dec 4 '10 at 22:13