Take the tour ×  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 $R(\Delta')$???  share|improve this question
add comment  1 Answer  up vote 4 down vote accepted  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.  share|improve this answer add comment  Your
Answer