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 $ad(E\_G)$? 



Is there any Lie algebra structure on the sheaf of sections of adjoint bundle
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.


