  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 $2$.
