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Help with Dirac trace algebra

1. Homework Statement
This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first question is: does it matter *which* gamma matrices? For example, one of my traces has
$$ \gamma_{\nu}\gamma^5\gamma^{\rho}\gamma^{\alpha}\gamma^{\sigma}\gamma^0\gamma^5\gamma^0\gamma^{\lambda}\gamma_{\mu}\gamma^{\beta} $$ Since each $$\gamma^5$$ is a product of 4 gamma matrices, altogether this would be a product of 17, which is odd. But some are $$\gamma^{\mu}$$, some $$\gamma^0$$ and some $$\gamma^5$$ and I'm just not sure what the rules are for this type of thing.

2. Homework Equations
$$Tr (ABC) = Tr (CAB) = Tr (BCA)$$
$$Tr (A +B) = Tr(A)+Tr(B)$$
$$Tr(aA) = aTr(A)$$

3. The Attempt at a Solution
I would think that this would, in fact, still be zero because there's an odd number of them. Otherwise I would probably use commutation/anticommutation relations to bring the $$gamma^0$$ and $$\gamma^5$$ together (respectively) to get unity, then just deal with the rest.
 
So does that mean that all the standard trace identies for gamma matrices (e.g. in Griffiths or Peskin Schroeder) hold for *any* gamma matrices, regardless of their index?
 

Orodruin

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*any* gamma matrices, regardless of their index?
As long as that index is 0, 1, 2, or 3. You will need to treat ##\gamma^5## differently.
 
Isn't the last one just a product of the first 4?
 

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