when a set of roots extend to a system of simple roots Given a set of roots in a root system, assume that the pairing of each two roots in this set is not positive. Then clearly the set gives a closed root subsystem. My question is, when this set extends to a system of simple roots in the original root system. Of course, it is not always true, for example taking closed subsystem of long roots in B_2. Is it true for any simply-laced case.  We are particularly interested in the exceptional type E_6.
 A: Let $\alpha_1,\ldots,\alpha_6$ be the simple roots of $E_6$, and suppose that $\alpha_3$ is the root corresponding to the trivalent node of the Dynkin diagram. Let $\theta$ be the highest (positive) root. Then $\{\alpha_1,\alpha_2,\alpha_4,\alpha_5,\alpha_6,-\theta\}$ is a system of simple roots for a root system of type $A_2 \times A_2 \times A_2$, so it clearly does not extend to a system of simple roots for $E_6$.
This example is coming from Borel-de Siebenthal theory, which basically says that the maximal rank sub-root systems of a root system are given by taking the extended Dynkin diagram of the Dynkin diagram, and deleting some node. The "affine node" of the extended Dynkin diagram corresponds to $-\theta$ the negative highest root.
(By the way, if you insist that the system of simple roots give an irreducible root system, then the answer changes in this case of $E_6$, because the only connected Dynkin diagrams we can get from the extended Dynkin diagram by deleting a node are of type $E_6$.)
EDIT: Let me mention a positive result, since you seem interested in that as well. Let $\Phi$ be a root system in a vector space $V$. Let $V'\subseteq V$ be a subspace spanned by a subset of roots. Set $\Phi' := \Phi\cap V'$, a sub-root system of $\Phi$. Let $S'$ be a system of simple roots for $\Phi'$. Then $S'$ can be extended to $S$, a system of simple roots for $\Phi$. This is for instance Bourbaki, "Lie groups and Lie algebras," Chapter VI, Section 1.7, Proposition 24.
