Krdytkyu

Let $X$ be an irreducible smooth projective variety over $mathbb{C}$.
Let $G$ be an affine algebraic group over $mathbb{C}$.
Let $p : E_G longrightarrow X$ be a holomorphic principal $G$-bundle on $X$. Let $ad(E_G) = E_G times^G mathfrak{g}$ be the adjoint vector bundle of $E_G$ associated to the adjoint representation $ad : G longrightarrow End(mathfrak{g})$ of $G$ on its Lie algebra $mathfrak{g}$. The fibers of $ad(E_G)$ are $mathbb{C}$-linearly isomorphic to $mathfrak{g}$.
Consider $ad(E_G)$ as a sheaf of $mathcal{O}_X$-modules on $X$.

Question: Is there any $mathcal{O}_X$-bilinear homomorphism
$[,] : ad(E_G)times ad(E_G) to ad(E_G)$ giving a Lie algebra structure on the sheaf $ad(E_G)$?

A principal $G$-bundle gives a monoidal functor from the category of representations of $G$ to the category of vector bundles. In particular, it takes the morphism
$$
[-,-] colon mathfrak{g} otimes mathfrak{g} to mathfrak{g}
$$
of $G$-representations (for the adjoint action) to a morphism of vector bundles
$$
[-,-] colon ad(E_G) otimes ad(E_G) to ad(E_G).
$$
By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $ad(E_G)$ with a Lie algebra structure.

Yes. It boils down to natural isomorphism $ad(E_G) otimes ad(E_G) simeq E_G times^G (mathfrak{g}otimes mathfrak{g})$ which allows you to compose tensor product of sections with the bracket on $mathfrak{g}otimes mathfrak{g}$.