Vector-matrix algebras and their isomorphisms

Starting from a formed algebra, an algebra equipped with a bilinear form satisfying certain conditions, we construct its vector-matrix algebra, generalizing the Zorn vector-matrix construction of the split octonions. We explore how different varieties of formed algebras induce various properties in their corresponding vector-matrix algebras. We identify when two formed algebras of a particular variety generate isomorphic vector-matrix algebras. The existence of such isomorphisms can be reduced to relationships between the underlying formed algebras, and for this we give some explicit examples.