LDB division algebras

An LDB division algebra is a triple $(A,\star,\bullet)$ in which $\star$ and $\bullet$ are regular bilinear laws on the finite-dimensional non-zero vector space $A$ such that $x \star (x \bullet y)$ is a scalar multiple of $y$ for all vectors $x$ and $y$ of $A$. This algebraic structure has been recently discovered in the study of the critical case in Meshulam and \v Semrl's estimate of the minimal rank in non-reflexive operator spaces. In this article, we obtain a constructive description of all LDB division algebras over an arbitrary field together with a reduction of the isotopy problem to the similarity problem for specific types of quadratic forms over the given field. In particular, it is shown that the dimension of an LDB division algebra is always a power of $2$, and that it belongs to $\{1,2,4,8\}$ if the characteristic of the underlying field is not $2$.