Why does Σ¹ = 2iγ²γ³ from the Levi-Civita commutator formula?

[vertices]

Sorry for spamming the forums, but one last question for today!

If

$$\Sigma^k=\frac{i}{2} \epsilon_{kij} [\gamma^i , \gamma^j]$$

where [A,B]=AB-BA

Why does $${\Sigma^1=2i \gamma^2\gamma^3$$ (that's what my notes say, anyway)

I think it should equal:

$$\Sigma^1=\frac{i}{2}\epsilon_{123}[\gamma^2,\gamma^3] + \frac{i}{2}\epsilon_{132}[\gamma^3,\gamma^2]$$
$$= \frac{i}{2}(+1)}[\gamma^2,\gamma^3] + \frac{i}{2}(-1)(-[\gamma^2,\gamma^3])$$
$$=i[\gamma^2,\gamma^3]$$

 

[vertices]

Disregard that question - I've just worked it out. For anyone who's interested: it's because

$$=\gamma^\mu\gamma^\nu = -\gamma^\nu\gamma^\mu$$

 

[ansgar]

Disregard that question - I've just worked it out. For anyone who's interested: it's because

$$=\gamma^\mu\gamma^\nu = -\gamma^\nu\gamma^\mu$$

that is not true, it is only true for the "spatial" components, for the time component: gamma^0 gamma^0 = gamma^0 gamma^0 = 1