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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.