Building positive roots from simple roots

[alexgs]

Hi, I've got two related questions.

You can decompose a (semisimple) lie algebra into root spaces, each of which are 1-dimensional. If X has root a and Y has root b then [X,Y] has root a+b. If the root space of a+b is not zero (i.e. there is a root a+b) then is it possible for [X,Y] to still be 0?

I ask because in Georgi on p.107 he has an "inductive" method of building all the positive roots from the simple ones where, if X,Y,... are the simple roots you just form all the brackets of them and check which ones are roots.

For instance, if X and Y are simple you want to see if there are any positive roots a+b. So you form [X,Y] and then check to see if it's a root. But isn't it possible that [X,Y]=0 even if the root space of a+b is not empty? In this case, wouldn't the "algorithm" miss the root a+b?

Thanks in advance.
Alex

 

[fresh_42]

Here's the method explained (for $G_2$):
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/