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I'm currently reading a book on particle physics, which tells me this about SU(3):

"...The generators may be taken to be any 3x3-1=8 linearly independent traceless matrices. Since it possible to have only two of these traceless matrices diagonal, this is the maximum number of commuting generators..."

These few lines make close to no sense to me. I have some questions:

1. Why 3x3-1? A 3x3 matrix has 9 entries, so that might be the 9, and the 1 must somehow come from the unitarian and determinant=1 conditions. But how?

2. Why do they need to be traceless?

3. Most importantly: Why do you have a maximum of 2 matrices diagonal, and what does this have to do with the number of commuting generators?

Then he goes on being sensible again, but this I don't get.

If anyone can help me through these questions, I'd be very grateful.

Jacob