I want to check the following statement:
Suppose $G\rightarrow P \rightarrow M$ be a principal bundle, and $E=P\times_{\rho} V$ be an associated vector bundle defined by representation $\rho$, then a section $\alpha$ of the adjoint bundle $Ad(P)$ induces a canonical endomorphism of $E$.
Section $\alpha$ attaches a point $x$ in $M$ an element in $GL(\mathfrak{g})$, how can I use this to create an endomorphism of $E$? Since the representation $V$ has not to be a subspace of $\mathfrak{g}$, I don’t see how to use $\alpha(x)$.
This is not quite correct. The adjoint bundle $Ad(P)$ is the quotient space $P\times_{Ad}\mathfrak{g}$. We can express any section $\alpha:M\to Ad(P)$ as $$ \alpha(x)=[p,\tilde\alpha(p)]_{Ad}, $$ where $p$ is any element of the fiber in $P$ above $x$, and $\tilde{\alpha}$ is a function from $P$ to $\mathfrak{g}$ satisfying $\tilde\alpha(p\cdot g) = Ad_{g^{-1}}(\tilde\alpha(p))$. Here $[p,\xi]_{Ad}\in Ad(P)$ denotes the equivalence class of $(p,\xi)\in P\times \mathfrak{g}$.
The section $\alpha$ then acts on any associated bundle $P\times_\rho V$ as $$ \alpha(x)\cdot [p,v]_\rho = [p,\dot\rho(\tilde\alpha(p))v]_\rho, $$ where $\dot\rho:\mathfrak{g}\to L(V)$ denotes the derivative of the representation $\rho:G\to GL(V)$. It follows from the compatibility condition for $\tilde\alpha$, plus the fact that $\dot\rho(Ad_{g^{-1}}\xi) =\rho(g^{-1})\dot\rho(\xi)\rho(g)$, that this definition is well-defined.