It is extremely well-known that $H^*(BQ_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and $|\gamma|=4$. I was wondering if anyone can given an explicit reference showing that $c_2(\sigma)$ generates $H^4(BQ_8)\cong \mathbb{Z}_8$, where $\sigma: Q_8 \hookrightarrow SU(2)$ is the standard representation (i.e., $c_2(\sigma)=c_2(E_\sigma)$, $E_\sigma= S^\infty\times_{Q_8}\mathbb{C}^2)$.

Any help would be greatly appreciated, as I would like to avoid the following reasoning if possible:

$E_\sigma$ is the pullback of the canonical quaternionic line bundle $E=S^\infty\times_{SU(2)}\mathbb{C}^2\rightarrow \mathbb{H}P^\infty$ by the projection $S^3/Q_8\hookrightarrow BQ_8 \stackrel{\pi}{\rightarrow}\mathbb{H}P^\infty$, and Leray-Hirsch gives $H^4(\mathbb{H}P^\infty;\mathbb{Z}_2)\cong H^4(BQ_8;\mathbb{Z}_2)\cong\mathbb{Z}_2$. Looking at $H^4(\mathbb{H}P^\infty;\mathbb{Z})\cong \mathbb{Z}\rightarrow H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ then shows that $c_2(\rho)=\pi^*(c_2(E))$ must therefore represent an odd number in $\mathbb{Z}_8$.