  We prove that in any strictly convex symmetric cone $\Omega$ there exists a
non empty locus where the WDVV equation is satisfied (i.e. there exists a
hyperplane being a Frobenius manifold). This result holds over any real
division algebra (with a restriction to the rank 3 case if we consider the
field $\mathbb{O}$) but also on their linear combinations. This theorem holds
as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz
symmetric cone of Anti-de-Sitter type. Our statement can be considered as a
generalisation of a result by Ferapontov--Kruglikov--Novikov and Mokhov. Our
construction is achieved by merging two different approaches: an
algebraic/geometric one and the analytic approach given by Calabi in his
investigations on the Monge--Amp\`ere equation for the case of affine
hyperspheres.
