Back in the 19th century, when people had been experimenting with determinants a lot, they might have interpreted the above definition of $B\times C$ in terms of quaternions. If $i$, $j$, and $k$ denote basis elements of $\mathbb H$ and 
$${\mathbf x}=x_1i+x_2j+x_3k,$$
$${\mathbf y}=y_1i+y_2j+y_3k\quad$$
are pure imaginary elements of $\mathbb H$, then the vector part of the Hamilton product 
$\mathbf{xy}$ is equal to the determinant

$$\Im(\mathbf{xy})=\Im(\mathbf{x})\times \Im(\mathbf{y})=\det 
\begin{vmatrix}
  i  &   j  &   k    \\\\
      x_1 & x_2 & x_3  \\\\
      y_1 & y_2 & y_3\\\\   
\end{vmatrix}.$$


There is a [note][1] by Sir Arthur Cayley  where he introduces the notion of a quaternion determinant. He mentions several identities of the form 
 
$$ \det 
\begin{vmatrix}
 {\mathbf  x} & {\mathbf  x}  \\\\
  {\mathbf  y} & {\mathbf  y}     
     \\\\   
\end{vmatrix}
    = 
  -2\det 
\begin{vmatrix}
  i  &   j  &   k    \\\\
      x_1 & x_2 & x_3  \\\\
      y_1 & y_2 & y_3\\\\   
\end{vmatrix}
$$ 
and 
$$ \det 
\begin{vmatrix}
 {\mathbf  x } & {\mathbf  x } & {\mathbf  x }   \\\\
 {\mathbf  y } & {\mathbf  y } & {\mathbf  y }      \\\\
  {\mathbf  z } & {\mathbf  z } & {\mathbf z }   \\\\   
\end{vmatrix}
    = 
  -2\det 
\begin{vmatrix}
{3} &  i  &  j  &   k   \\\\
   x_0 &   x_1 & x_2 & x_3  \\\\
   y_0 &  y_1 & y_2 & y_3\\\\   
   z_0 &  z_1 & z_2 & z_3\\\\   
 \end{vmatrix}
$$
where  $\mathbf x$, $\mathbf y$, $\mathbf z$ are arbitrary quaternions
$${\mathbf x}=x_0+x_1i+x_2j+x_3k, \mbox{ etc.}$$




  [1]: http://www.informaworld.com/smpp/content~db=all~content=a910858261