  We study the second fundamental form of the Siegel metric in $\mathcal A_5$
restricted to the locus of intermediate Jacobians of cubic threefolds. We prove
that the image of this second fundamental form, which is known to be
non-trivial, is contained in the kernel of a suitable multiplication map. Some
ingredients are: the conic bundle structure of cubic threefolds, Prym theory,
Gaussian maps and Jacobian ideals.
