Problem with SU(3) generators's trace

[Einj]

Hi everyone. I'm not sure this is the correct section for this topic and if not my apologiez.
I'm studying SU(3) and my professor wrote down the following equality:

$$Tr\left(\left[ T^a_8,T^b_8\right] T^c_8\right)=i\frac{3}{2}f^{abc}$$

where Ts are generators of the adjoint representation. I'm not sure this relation is correct and I would like to have your opinion. The Dynkin index of the adjoint representation is 3 so:

$$Tr\left(T^a_8T^b_8\right)=3\delta^{ab}$$

Now, my reasoning is:

$$Tr\left(\left[T^a_8,T^b_8\right]\right)=if^{abd}Tr(T^d_8T^c_8)=if^{abd}3\delta^{dc}=3if^{abc}$$

The difference is just a 1/2 factor but I would like to know if I'm doing something wrong.

Thanks everybody

 

[andert]

The index for SU(3) should be 3 in the adjoint representaion (and N generally). You should probably ask him to clarify.