the first pontryagin class of a quaternionic line bundle over a CW-complex is zero if and only if the quaternionic line bundle is trivial or not?
Let $\xi^\mathbb{H}$ be a given quaternionic line bundle. Is there any method to see whether $p_1(\xi^{\mathbb{H}})=0$? Could you give references?
Quaternionic line bundles are classified by maps to $B GL_1(\mathbb{H}) \cong BSp(1)$. This has cohomology a polynomial algebra
$$H^{*}(BSp(1)), \mathbb{Z}) \cong \mathbb{Z}[p_1]$$
on the Pontryagin class $p_1 \in H^4$.
For complex line bundle and chern class, I obtained that the first chern class is zero if and only if the complex line bundle is trivial. This is obtained by $$ c_1(\xi^\mathbb{C})=e((\xi^\mathbb{C})_\mathbb{R})=o_2((\xi^\mathbb{C})_\mathbb{R}). $$ By page 140, 143, 158, Charachateristic class, Milnor and Stasheff, we obtain that the first chern class is zero if and only if the complex line bundle is trivial.
But for quaternionic line bundle $\xi^{\mathbb{H}}$, page 174, Charachateristic class, Milnor and Stasheff, $$ p_1(\xi^\mathbb{H})=p_1((\xi^\mathbb{H})_\mathbb{R})=-c_{2}((\xi^\mathbb{H})_\mathbb{R}\otimes\mathbb{C}). $$ I do not know how to do.