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 $R(\Delta')$???



Parabolic Subalgebra
Jim gave me the answer!
It is false in general!


Thanks.


