The space of two-dimensional vectors over a four-dimensional division algebra over $F_2$

Let $A$ be a non-associative division algebra over a field $F$.

Let $A$ act on the space $A^2$ by left multiplication.

For nonzero elements $v, v'$ of $A^2$ we ask when the subspaces $Av$ and $Av'$ coincide.

The paper gives an answer in the case where $A$ is four-dimensional and $F$ is the field of order 2.

In this case $A$ is known to be isotopic to either of two algebras, system $V$ and system $W$ of Knuth.

For each of these two algebras we give an explicit solution of the equation $Av=Av'$.

The result is stated for any four-dimensional algebra $A$ defined over an arbitrary base field $F$ and equipped with the multiplication rule of system $V$ or system $W$.