orbit. This space is classified into a special case $(n,m)=(4,2)$ of the irreducible reduced prehomogeneous vector spaces (20) $(\SO_n\times \GL_m,M_{n,m})$ in [@SK Theorem 24 in § 3]. Its split $F$-form $F^2\otimes F^2\otimes F^2$ was introduced in [@WY] as the $D_4$ case from the viewpoint of field extensions, and its $\Z$-structure $\Z^2\otimes \Z^2\otimes \Z^2$ is well-known as Bhargava’s cube [@Bhargava]. Its $F$-forms were classified in \[Saito, Theorem 2.11 (3)\]. The $F$-rational orbits of the non-split $F$-forms $\GL_2(E)\times \GL_2(F)$ and $\GL_2(E')$, where $E$ (resp. $E'$) is a quadratic (resp. cubic) extension of $F$, were studied in [@KY] and [@GS]. In addition, Kable and Yukie studied their zeta functions and density theorems in [@KY1], [@Yukie] and so on. Besides, Taniguchi [@Taniguchi1; @Taniguchi2] studied another type non-split $F$-form, which is a pair of quaternoin algebras. In this section, we review basic properties of the spaces of pairs of quaternion algebras, which include the split case $F^2\otimes F^2\otimes F^2\cong M_2\oplus M_2$.

### Setup

Let $D$ denote a quaternion algebra over $F$. When $D$ is not division, we may identify $D$ with $M_2(F)$, and a standard $F$-involution $\iota$ on $D$ can be chosen as $x^\iota:=\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)\,{}^t\!x \left(\begin{smallmatrix} 0 & 