So, suppose given a root system in a euclidean space , which arises from a semisimple Lie algebra and a Cartan subalgebra as before. The first goal of this post is to discuss the “splitting”

(disjoint union) in a particular way, into positive and negative roots, and the basis decomposition into simple roots. Here .

To do this, choose such that for . Then define to be those roots with and those with . This was easy. We talked about positive and negative roots before using a real-valued linear functional, which here is given by an inner product anyway.

**Bases **

OK. Next, I claim it is possible to choose a linearly independent set such that every root is a combination

with all the or all the .

Then will be called a **base**. It is not unique, but I will show how to construct this below. (more…)