# SU(3) question

1. Dec 22, 2007

### ak99

Hi I am teaching myself lie algebras from a freely available book, can anyone familiar with the subject help me with a simple question.

1. The problem statement, all variables and given/known data
The problem is problem #2 of chapter 2 of the book Semi-Simple Lie Algebras and their Representations by Robert Cahn, the book is freely available at http://www-physics.lbl.gov/~rncahn/book.html [Broken].

It is basically an identity involving the sum of products of entries of Gell-Mann matrices:

$$\sum_{i} \lambda^{i}_{ab} \lambda^{i}_{cd} = - 1/3 \sum_{i} \lambda^{i}_{ad} \lambda^{i}_{cb} + \frac{16}{9} \delta_{ad}\delta_{cb}$$
, my question is that is there any way to prove it other than brute force? If not how did anyone figure out to single out this identity?

2. Relevant equations

3. The attempt at a solution

Simple brute force calculation can be done.

Last edited by a moderator: May 3, 2017