This article says that for an Ehresmann connection $H \subseteq TE$ of a vector bundle $E \to M$ to be linear (and so to define a connection in the usual sense as a linear covariant derivative operator) all you need is
$$D(S_\lambda)_e(H_e) = H_{\lambda e}$$ for $\lambda \in \mathbb{R}$ and $e \in E$, where $S_\lambda : E \to E$ is the multiplication by $\lambda$ map. But to me it seems like we also need $$D\sigma_{(e,e')} (H_e \oplus H_e') = H_{e + e'}$$ where $\sigma : E \oplus E \to E$ is the addition map and $e,e'$ lie on the same fiber. Can this somehow be deduced from the previous property, or is the claim in the article mistaken?