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.



when a set of roots extend to a system of simple roots
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](https://en.wikipedia.org/wiki/Borel%E2%80%93de_Siebenthal_theory), which basically says that the maximal rank sub-root systems of a root system are given by taking the [extended Dynkin diagram](https://en.wikipedia.org/wiki/Dynkin_diagram#Affine_Dynkin_diagrams) 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.


