Question: Just a d\*mb question on Lie algebras:

Given a Dynkin diagram of a root system (or a Cartan Matrix), how do I know which combination of simple roots are roots?

Eg. Consider the root system of G\_2, let a be the short root and b be the long one, it is clear that a, b, b+a, b+2a, b+3a are positive roots. But it is not clear to me that 2b+3a is a root just from the Dynkin diagram.

Answer: 
There is also a succinct description of the algorithm in he very last paragraph of §11.1 in page 56 of Humphrey's *Introduction to Lie algebras and representation theory*.

Here's the oblgatory [Google Books link](http://books.google.com/books?id=mSD4DvUFa6QC&lpg=PP1&dq=Humphreys%20Lie%20algebras&client=safari&pg=PA56#v=onepage&q=Coxeter%20graphs%20and%20Dynking%20diagrams&f=false).

I remember writing for fun a Fortran program (later translated to C) to compute the roots given a Cartan matrix. The algorithm has the virtue that if you input the Cartan matrix of an affine Kac-Moody algebra, say, it will still compute the roots level by level, except that, of course, it will never stop running :)
Just a related point for hashing out arguments quickly:

If you ever find yourself in the situation where you want to rule out some vector being a root in a hurry, you can check it against the other roots you do know to see if you get a Cartan integer out of their dot products.