# On Finding Lie Algebra Representations

antibrane
What I am trying to do is start with a Dynkin diagram for a semi-simple Lie algebra, and construct the generators of the algebra in matrix form. To do this with su(3) I found the root vectors and wrote out the commutation relations in the Cartan-Weyl basis. This gave me the structure constants of the algebra and then I was able to write down the matrices corresponding to the regular (adjoint) representation.

What I am confused about (assuming the above is a correct method) is how to find different matrix representations than this. The representation I am familiar with for su(3) consists of the "Gell-Man matrices" which are 3 dimensional and obviously my matrices from my construction are 8 dimensional.

How would I find different representations starting from the commutation relations? I know how to find them starting with the Lie Group. Let me know if I need to clarify something--thanks a lot.