# Chern class of a principal $G$ bundle for a compact Lie group $G$

This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here.

Let $$G$$ be a compact Lie group and $$P\rightarrow M$$ be a principal $$G$$ bundle. We want to associate Chern classes for this bundle.

This article says in its second page that "A compact Lie group $$G$$ may be shown (with some work, using the theory of compact operators) to be none other than a closed subgroup of the Unitary group $$U(n)$$ of matrices
for some $$n$$". Thus, for structure group $$G$$ of $$P(M,G)$$ there exists an embedding $$G\hookrightarrow U(n)$$.

See the embedding $$G\hookrightarrow U(n)$$ as an action of $$G$$ on $$U(n)$$. Given a manifold $$P'$$ and an action of $$G$$ on $$P'$$ there is a notion of what is called an associated bundle whose fibres are $$P'$$. In case $$P'$$ is a Lie group $$H$$, we get a principal $$H$$ bundle. This $$G$$ action on $$U(n)$$ gives a principal $$U(n)$$ bundle $$Q\rightarrow M$$.

For this principal $$U(n)$$ bundle $$Q\rightarrow M$$, we fix a connection $$\Gamma$$ with curvature $$\Omega$$. We then get what is called Weil homomorphism $$W:I(U(n))\rightarrow H^*(M,\mathbb{R})$$.

We choose elements $$f_k:\mathfrak{u}(n,\mathbb{C})\rightarrow \mathbb{R}$$ from $$I(U(n))$$ such that $$\text{det}\left(\lambda I-\frac{1}{2\pi\sqrt{-1}}X\right)=\sum_{k=0}^n f_k(X)\lambda^{n-k}$$ The images $$W(f_k)\in H^{2k}(M,\mathbb{R})$$ are called the $$k$$-th Chern classes of $$Q\rightarrow M$$.

How do we define Chern classes of the principal $$G$$ bundle $$P\rightarrow M$$ that we have started with?

Is the $$k$$-th Chern class of $$G$$ bundle $$P(M,G)$$ the $$k$$-th Chern class of the $$U(n)$$ bundle $$Q(M,U(n))$$?

Does this depend on the embedding $$G\hookrightarrow U(n)$$ we have chosen?

In case it is dependent on the choice of $$G\hookrightarrow U(n)$$, what invariants of $$P(M,G)$$ does these give?