I am just starting to look at this concept in particle physics, and wondering if someone could clear something up for me. It concerns irreducible representations and the Cartan-Weyl basis. (I'm sorry if it sounds really ignorant but I'm stuck!)

In physics we almost always work with unitary representations, and since these can be expressed as the direct sum of irreducible representations we basically always work with irreducible representations of our algebras.

Now, I understand that any simple Lie algebra may be put into its Cartan-Weyl basis. But what I am wondering is this: suppose I make an irreducible representation of my algebra. Can I ask: is the Cartan-Weyl basis designed to only be used with irreducible representations? If not, if I start off with an irreducible representation and then put it into Cartan-Weyl form, can I guarantee that it will still constitute an irreducible representation?

Any help would be very happily received.