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.
For instance, there is a note 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 $i$, $j$, and $k$ are basis elements of $\mathbb H$ and $\mathbf x$, $\mathbf y$, $\mathbf z$ are arbitrary quaternions: $${\mathbf x}=x_0+x_1i+x_2j+x_3k, \mbox{ etc.}$$