Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class $$ w_1(\xi)=0 $$ if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(n)$.

**Question 1:** Let $\xi^\mathbb{C}$ be a complex vector bundle of dimension $n$. Then the first Chern class
$$
c_1(\xi^\mathbb{C})=0
$$
if and only if the structure group of $\xi^\mathbb{C}$ can be reduced to $SU(n)$? Is it true or false? Any references?

**Question 2:** Let $\xi^\mathbb{H}$ be a quaternion vector bundle of dimension $n$. Then the first Pontrjagin class
$$
p_1(\xi^\mathbb{C})=0
$$
if and only if the structure group of $\xi^\mathbb{H}$ can be reduced to $SSp(\mathbb{H},n):=\{A\in Sp(n)\mid \text{Det}(A)=1\}$? Is it true or false? Any references?

I have a further question characteristic classes of a covering space with symmetric group action

can be reducedto $SU(n,\mathbb C$ or $\mathbb H)$, in which case the answers are "true". This is standard characteristic class stuff, as in Milnor's book or Husemoller's, not research-level. $\endgroup$isno nontrivial normal subgroup $\mathrm{SSp}(\mathbb{H},n)\subset \mathrm{Sp}(n)$ (other than the center, which is discrete) because the group $\mathrm{Sp}(n)$ is simple. Thus, there is no interpretation of $p_1(\xi)=0$ as a condition for reducing the structure group of a quaternionic vector bundle $\xi$ to such a subgroup. $\endgroup$2more comments