$\mathcal{E} \nabla^4 R^4$ type invariants and their gradient expansion

We analyse the constraints from supersymmetry on $\nabla^4 R^4$ type corrections to the effective action in $\mathcal{N}=2$ supergravity in eight dimensions. We prove that there are two classes of invariants that descend respectively from type IIA and type IIB supergravity. We determine the first class as $d$-closed superforms in superspace in eight dimensions, whereas we obtain the second class by dimensional reduction down to four dimensions, in which there is a single class of invariants transforming in the next to minimal unitary representation of $E_{7(7)}$.