Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb C^{248}$. This vector bundle has an $E_8$ characteristic classes $\lambda (V)\in H^4(X,\mathbb Z)$ obtained by pulling back the generator $\lambda \in H^4(BE_8,\mathbb Z)$ and it has a second Chern class $c_2(V)$ given by pulling back the generator $c_2\in H^4(BSU(248),\mathbb Z)$.

I am looking for a reference for the following fact: $$\lambda (V)=\frac{c_2(V)}{60}.$$

There is the map $B\rho^* : H^4(BSU(248),\mathbb Z)\rightarrow H^4(BE_8, \mathbb Z)$. Since both groups are canonically isomorphic to $\mathbb Z$, the map is determined by a single integer, which is the Dynkin index of $E_8$ and has been computed to be $60$. The above fact essentially follows from this. I am writing a paper and would prefer to just state the fact and then point the reader to whoever first presented a thorough argument filling in all the details. I have seen it mentioned as a footnote on page 68 in this paper by Friedman, Morgan, and Witten and some of the details regarding the Dynkin index are discussed in this paper by Totaro.