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.