Suppose $V$ is a vector bundle with structure group $SO(3)$, and suppose that it can be lifted to a $\text{Spin}(3) = SU(2)$ bundle (i.e. $w_2(V) = 0$). Let us call the lifted bundle $E$. Then it is stated on page 42 in *The Geometry of Four-Manifolds* by Donaldson and Kronheimer that we have the relation $p_1(V) = -4c_2(E)$. My question is:

How does one show this?

More generally, how does one compute the effect of going over to a lifted bundle on the characteristic classes? Generally one has $p_1(E) = c_1^2(E) - 2c_2(E)$. In our case the first term drops out, so that the claim in the book can also be written as $p_1(V) = 2p_1(E)$. This factor $2$ undoubtedly somehow comes from the covering homomorphism $SU(2) \rightarrow SO(3)$ which is 2 : 1, but how?

Probably related: on the same page he defines at the bottom for the so-called instanton number $\kappa = \frac1{8\pi^2}\int_M\text{Tr}(F^2)$, and then claims that this is $\kappa = c_2$ for $SU(r)$ bundles $E$ and $\kappa = -\frac14p_1$ for $SO(r)$ bundles $V$. There again is that factor 4; again, from the formula $p_1(E) = c_1^2(E) - 2c_2(E) = \left[\frac{-1}{4\pi^2}\text{Tr}(F^2)\right]$ one would expect this to be 2. I can see that this factor is chosen so that if one lifts the bundle that $\kappa$ does not change, but on the other hand, not every bundle is liftable, and for bundles which have both Chern classes and Pontryagin classes (such as complex $SU(r)$ bundles), I would expect that one would want the two formulae to give the same answer. As it is, they don't.

(I guess the real problem is I haven't managed to find any readable sources on $SO(r)$-bundles in the context of gauge theories. For special unitary groups there are sources in abundance, but for special orthogonal they are a lot harder to find.)