Relating the Chern classes of vector bundles with structure groups $G_1$, $G_2$, and $G_1\times G_2$

Question

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 [1], but I don't know if these observations have anything to say about my question.

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Reference:

1. Page 22 in Cohen (1998), The Topology of Fiber Bundles Lecture Notes (http://math.stanford.edu/~ralph/fiber.pdf)

Answer 1

If $E \to X$ is a complex vector bundle with structure group $G=U(1)\times SU(n-1) \subset U(n)$ there exists complex line bundle $E_1$ (with structure group $U(1)$) and a complex vector bundle $E_2$ with structure group $SU(n-1)$ such that $E$ is isomorphic to $E_1 \oplus E_2$.

This makes the Chern classes of $E$ computable in the terms of chern classes of $E_1$ and $E_2$.

For the statement, consider the $G$ principle bundle $P\to X$ associated to $E$. The proection \begin{align*} pr_1:G \to U(1) \end{align*} induces a $U(1)$ principle bundle $P_1 := (P \times U(1))/G$. Where the right action of $G$ on $P\times U(1)$ is given by \begin{align*} (p,z)\cdot g:= (p\cdot g, pr_1(g)^{-1}\cdot z). \end{align*}

Analogously $pr_2:G \to SU(n-1)$ defines a $SU(n-1)$ principle bundle $P_2$. There is an ismorphism of $G$ principle bundles $P \overset{\sim}{\to} P_1 \times P_2$. Take $E_1,E_2$ to be the associated vector bundles respectively.

This corresponds to a map of classifying spaces $B(G_1\times G_2) \to BG_1 \times BG_2$ which is homotopy inverse to $BG_1\times BG_2 \to B(G_1 \times G_2)$.