The three-dimensional bundle underlying a quaternionic line bundle Consider a principal $Sp(1)$-bundle $P$ over a compact space $X$, i.e. a quaternionic line bundle. We can push $P$ forward along the double cover $Sp(1) \cong SU(2) \to SO(3)$ to get a three-dimensional real oriented vector bundle $E$ over $X$.
Is there a nice interpretation of $E$ in terms of $P$? For example, can we say something about its characteristic class $w_2(E)$ in terms of $p_1(P)$?
 A: If the $SU_2$-bundle $P\rightarrow X$ is classified by a map $f:X\rightarrow BSU_2$, then the pushforward $E$ is classified by the composite $X\xrightarrow{f} BSU_2\xrightarrow{B\theta} BSO_3$, where $B\theta$ is the classifying map of the projection $\theta:SU_2\rightarrow SO_3$. Thus $P\cong f^*ESU_2$ and $E\cong(B\theta\circ f)^*ESO_3\cong f^*(B\theta^*ESO_3)$. Since $H^2(BSU_2;\mathbb{Z}_2)=0$ we get $w_2(B\theta^*ESO_3)=0$, so
$$w_2(E)=f^*w_2(B\theta^*ESO_3)=0.$$
On the other hand the Pontryagin class $p_1(E)$ is a more interesting computation. We'll proceed again by means of the universal example.
Notice that the double cover $SU_2\cong Spin_3\rightarrow SO_3$ is exactly the adjoint representation of $SU_2$. Thus if $Q$ is the real vector bundle associated to $B\theta^*ESO_3$, then we can write $Q\cong ESU_2\times_{Ad}\mathfrak{su}_2$, where $\mathfrak{su}_2$ is the Lie algebra of $SU_2$. Now $p_1(B\theta^*ESO_3)=p_1(Q)$, and $p_1(Q)=-c_2(Q^\mathbb{C})$ by definition, where $Q^\mathbb{C}$ is the complexification of $Q$. By the above this is $Q^\mathbb{C}\cong ESU_2\times_{Ad}(\mathfrak{su}_2)^\mathbb{C}$ and we'll return to this is a second. First observe that the Chern classes are stable, so if $\epsilon$ is a trivial complex line bundle, then $c_2(Q)=c_2(Q\oplus\epsilon)$.
Now, the complexification of $\mathfrak{su}_2$ is the complex Lie algebra $\mathfrak{sl}(2;\mathbb{C})$ of traceless $2\times2$ complex matrices, and there is an isomorphism $\mathfrak{sl}(2;\mathbb{C})\oplus\mathbb{C}\cong\mathfrak{gl}(2;\mathbb{C})\cong Hom(\mathbb{C}^2,\mathbb{C}^2)$, where the trivial $\mathbb{C}$ factor corresponds to the central subgroup of $\mathfrak{gl}(2;\mathbb{C})$ generated by complex multiples of the identity. This gives us
$$Q^\mathbb{C}\cong ESU_2\times_{Ad}\mathfrak{sl}(2;\mathbb{C}),\qquad\qquad Q^\mathbb{C}\oplus\epsilon\cong Hom(EU_2,EU_2)|_{BSU_2}\cong \overline{EU_2}\otimes EU_2|_{BSU_2}$$
where we identify the conjugate and dual bundles of $EU_2$ with the aid of a suitable Hermitian metric.
The point is that we can compute $c_2(Q^\mathbb{C})$ by computing $c_2(\overline{EU_2}\otimes EU_2)$ and resticting from $BU_2$ to $BSU_2$, and this is not so difficult to do. Indeed, introduce a formal factorisation of the total Chern class by writing $c(EU_2)=\prod^2_{i=1}(1+x_i)$. Then $c(\overline{EU_2})=\prod^2_{i=1}(1-x_i)$ and
$$c(\overline{EU_2}\otimes EU_2)=\prod_{i,j=1,2}(1+(x_i-x_j))=\prod_{1\leq j<i\leq 2}(1-(x_i-x_j)^2)=1-(x_1+x_2)^2+4x_1x_2.$$
The formulas here are lifted straight from Hirzebruch's book Topological Methods in Algebraic Geometry, Th. 4.3.3, pg. 64. In any case this gives us $c_1(\overline{EU_2}\otimes EU_2)=0$ and $c_2(\overline{EU_2}\otimes EU_2)=-c_1^2+4c_2$. Pulling this back to $BSU_2$ kills $c_1^2$ and we obtain
$$p_1(Q)=-c_2(Q^\mathbb{C})=-c_2(Q^\mathbb{C}\oplus\epsilon)=-4c_2.$$
Finally returning to the $SO_3$bundle $E\rightarrow X$ we can compute
$$p_1(E)=f^*p_1(Q)=-4\cdot f^*c_2=-4\cdot c_2(P)$$
where of course $P\rightarrow X$ is the $SU_2$-bundle we started with.
