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 $ad(E_G)$?