Normal sections and direct product decompositions

In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s : Y --> X of an arrow f : X --> Y produces a direct product decomposition of the form X similar or equal to Y x W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories