Metabelian associative algebras

Metabelian algebras are introduced and it is shown that an algebra $A$ is metabelian if and only if $A$ is a nilpotent algebra having the index of nilpotency at most $3$, i.e. $x y z t = 0$, for all $x$, $y$, $z$, $t \in A$. We prove that the It\^{o}'s theorem for groups remains valid for associative algebras. A structure theorem for metabelian algebras is given in terms of pure linear algebra tools and their classification from the view point of the extension problem is proven. Two border-line cases are worked out in detail: all metabelian algebras having the derived algebra of dimension $1$ (resp. codimension $1$) are explicitly described and classified. The algebras of the first family are parameterized by bilinear forms and classified by their homothetic relation. The algebras of the second family are parameterized by the set of all matrices $(X, Y, u) \in {\rm M}_{n}(k)^2 \times k^n$ satisfying $X^2 = Y^2 = 0$, $XY = YX$ and $Xu = Yu$.