  We provide a classification of developable cubic hypersurfaces in $\mathbb
P^4$. Using the correspondence between forms of degree $3$ on $\mathbb P^4$ and
Artinian Gorenstein $\mathbb K$-algebras, given by Macaulay-Matlis duality, we
describe the locus in ${\rm GOR}(1,5,5,1)$ corresponding to those algebras
which satisfy the Strong Lefschetz property.
