Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials $$ d_4^{p,m}:E_4^{p,3} \to E^{p+4,0} $$ for $m \in \{2,4,\ldots,2n\}$?