TITLE: first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
QUESTION [11 upvotes]: 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

REPLY [4 votes]: I think haohaizi was looking for the following standard facts:
$BSO(n)$ is the fiber of $Bdet : BO(n) \longrightarrow BO(1)$
and $Bdet = w_1$.  This uses $BO(1) = K(Z/2,1)$.
$BSU(n)$ is the fiber of $Bdet : BU(n) \longrightarrow BU(1)$
and $Bdet = c_1$.  This uses $BU(1) = K(Z,2)$.
There is no determinant function for quaternions;  they are not commutative.
As Sullivan suggests, $Sp(n)$ is already `special', i.e., oriented.  Further,
$Sp(1) = S^3$ is not a $K(Z,3)$, so the obstruction to orientability here is not a cohomology class.