For a given Jacobi-Jordan algebra A and a vector space V over a field k, a non-abelian cohomological type object ${\mathcal{H}}_A^2(V,A)$ is constructed: it classifies all Jacobi-Jordan algebras containing A as a subalgebra of codimension equal to ${\dim }_k(V)$. Any such algebra is isomorphic to a so-called unified product $A♮V$. 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 ${\mathrm{GL}}_k(V)⋊{\mathrm{Hom}}_k(V,A)$ and an Artin type theorem for Jacobi-Jordan algebra is proven.

对于给定的 Jacobi-Jordan 代数A和域k 上的向量空间V，非阿贝尔上同调类型对象${\mathcal{H}}_{一}^2(伏,一)$构造：它将所有包含A 的Jacobi-Jordan 代数分类为等于的余维子代数${\mathrm{暗淡}}_{克}(伏)$. 任何这样的代数同构于所谓的统一乘积 $一♮伏$. 此外，我们还介绍了双交叉（半直接、交叉或斜交叉）产品$一\bowtie 伏$与作为统一乘积的特例的两个 Jacobi-Jordan 代数相关联。提供了几个例子和应用：扩展的伽罗瓦群$一\subseteq 一\bowtie 伏$ 被描述为群的半直接积的子群 ${\mathrm{GL}}_{克}(伏)⋊{\mathrm{坎}}_{克}(伏,一)$ 证明了 Jacobi-Jordan 代数的 Artin 类型定理。