Let $E$ be a complex vector bundle whose structure group $G$ is a product: $G=G_1\times G_2$. If the fiber of $E$ is $n$-dimensional, then are the Chern classes of $E$ related to the Chern classes of vectors bundles with structure groups $G_1$ and $G_2$, also with $n$-dimensional fibers and over the same base?
In case it makes a difference, I'm especially interested in the case $G_1=U(1)$ and $G_2=SU(N)$, but the more general the better.
I'm aware of the Whitney sum formula (which expresses the Chern classes of a Whitney sum $E_1\oplus E_2$ of two vector bundles in terms of the Chern classes of the individual vector bundles $E_1$ and $E_2$), and I'm aware that the Whitney sum of $G_1$ and $G_2$ principal bundles is a $G_1\times G_2$ principal bundle , but I don't know if these observations have anything to say about my question.
- Page 22 in Cohen (1998), The Topology of Fiber Bundles Lecture Notes (http://math.stanford.edu/~ralph/fiber.pdf)