I'm reading the paper "Self-Duality Equations on a Riemann Surface" written by N.J.Hitchin in 1986. He is considering a principal $SO(3)$ bundle $P$ over a compact Riemann surface $X$ of genus $g\ge 2$ and he says we can always associate to $P$ a rank $2$ complex vector bundle $V$. To be able to do this we have to consider two cases:
- The second Stiefel-Whitney class $w_2(P)$ is equal to zero
- $w_2(P)\neq0$
In the first case the condition is equivalent to requiring that the principal $SO(3)$-bundle is spin and then there exist another principal bundle $\tilde{P}$ over $X$ whose structure group is Spin$(3)\cong SU(2)$ and to which we can associate a rank $2$ complex vector bundle $V$ with deg$(V):=\int_Xc_1(V)=0$. Here I have my first question:
Question 1
In this case the rank $2$ complex vector bundle $V$ associated to $\tilde{P}$ is given by $ad(\tilde{P}):=\tilde{P}\times_{SU(2)}\mathfrak{\mathfrak{su}(2)}$? Where $\mathfrak{su}(2)$ is the Lie algebra of $SU(2)$.
Moving on to the second case, Hitchin says that if $w_2(P)\neq0$ then there exist a $U(2)$-principal bundle $\bar{P}$ to which $P$ is associated via the homomorphism $U(2)/Z(U(2))\simeq SO(3)$. Associated to $\bar{P}$ is a rank $2$ complex vector bundle $V$ with odd degree.
Question 2
How can we associate the bundle $\bar{P}$ to $P$ following Hitchin?
I really do not understand the construction in the latter case. Thank you if anyone could help.