Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Consider the adjoint action of $ G$ on its Lie algebra, and let $(.,.)$ be a $G$-invariant inner product on $\mathfrak{g}.$ Let $\beta \in \mathfrak{g}.$
In some notes I'm reading, there is a 1-form defined by $$\alpha |_y := ([\beta,y],dy) \qquad y \in \mathfrak{g}$$
What is the definition of $dy$ in the expression of $\alpha|_y$? I would think it is $dy_1 \wedge ...\wedge dy_n$, where $y_i$ are the global coordinates functions on $\mathfrak{g}$, but since $\alpha$ is a 1-form, this is not possible.
If $ X \in \mathfrak{g}$, how to evaluate $\alpha |_y$ at the tangent vector $X_\mathfrak{g}(y):= \frac{d}{dt} \Bigg|_{t=0} e^{tX}.y = [X,y]. $
Edit: The author has defined a 1-form in more general context as follows,
Let $p: E \rightarrow M$ be a $G$- equivariant vector bundle provided with a covariant derivative $\nabla^E.$ Let $\nabla ^{p^*E}$ be the induced covariant derivative on the bundle $p^*E \rightarrow E$ and $\mathbf{y}$ be the tautological section of $p^*E$. Let $\alpha$ be the 1-form defined by $$\alpha=(\mathcal{L}(\beta) \mathbf{y}, \nabla^{p^*E} \mathbf{y})_E.$$
Any help would be very appreciated!