Algebraic constructions for Jacobi-Jordan algebras

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.