relating spinor and fundamental representation for $E_8$

Question

While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is $Tr(F^6)=\frac{1}{48}TrF^2TrF^4-\frac{1}{14400}(TrF^2)^3$, where trace is in adjoint representation.

I am able to prove this relation but while doing so, I have some identities which relates the spinor representation $128$ of $SO(16)$ to fundamental representation of $SO(16)$ which I must show but this is not working out.

The simplest one being $TrF^2=16trF^2$, where $Tr$ is in spinor representation $128$ and $tr$ is in fundamental representation. There are other relations showing the equality between $TrF^4$ and $tr(F^2)^2$ and $trF^4$. I am aware that spinor representation would be $\sigma_{ij}$ which is $128$ dimensional. While trying to prove these identities, I have noticed that if $F^2$ in the fundamental representation is diagonal with only two elements -1 and -1 and if $\sigma_{ij}^2=\frac{-I}{4}$ where $I$ is $128$ dimensional identity matrix then I can get the result. But I can not convince myself why it should be true.

Any details would be appreciated of how to prove it. The identities can be found in GSW chapter 13 last section (VOL.2).

This post imported from StackExchange Physics at 2014-08-01 20:15 (UCT), posted by SE-user user44895