Module derivations and cohomological splitting of adjoint bundles

Let $G$ be a finite loop space which satisfies the condition that the mod $p$ cohomology of the classifying space $BG$ is a polynomial algebra. We consider when the adjoint bundle associated with a $G$-bundle over $M$ splits on the mod $p$ cohomology as an algebra; that is, the mod $p$ cohomology algebra of the total space of the adjoint bundle is isomorphic to that of the product $G \times M$. In the case $p = 2$, an obstruction for the adjoint bundle to admit such a splitting is found in the Hochschild homology concerning the mod $2$ cohomologies of $BG$ and $M$ via a module derivation. Moreover the derivation tells us that such a splitting is not compatible with the Steenrod operations in general. As a consequence, we can show that the isomorphism class of an $SU(n)$-adjoint bundle over a $4$-dimensional CW complex coincides with the homotopy equivalence class of the bundle.