  For a given Jacobi-Jordan algebra $A$ and a vector space $V$ over a field
$k$, a non-abelian cohomological type object ${\mathcal H}^{2}_{A} \, (V, \,
A)$ is constructed: it classifies all Jacobi-Jordan algebras containing $A$ as
a subalgebra of codimension equal to ${\rm dim}_k (V)$. Any such algebra is
isomorphic to a so-called \emph{unified product} $A \, \natural \, V$.
Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed)
product $A \bowtie V$ associated to two Jacobi-Jordan algebras as a special
case of the unified product. Several examples and applications are provided:
the Galois group of the extension $A \subseteq A \bowtie V$ is described as a
subgroup of the semidirect product of groups ${\rm GL}_k (V) \rtimes {\rm
Hom}_k (V, \, A)$ and an Artin type theorem for Jacobi-Jordan algebra is
proven. The key tools for classifying supersolvable and flag Jacobi-Jordan
algebras are introduced.
