Let $P$ be a principal $SU(2)$ bundle over base $B$, $P'$ be the principal $SO(3)$ bundle obtained by quotient $\{1,-1\}$.
How do we show the first Pontryagin class of $P'\times \mathbb{R}^3$ equals 4 times the second Chern class of $P\times\mathbb{C}^2$ ?