  We rewrite the Born-Infeld Lagrangian, which is originally given by the
determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the
field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4
\cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$
are given by the linear combination of $g$ and $F$, it is found, based on the
representation matrix for the multiplication operator of the Cayley-Dickson
algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter,
where distinguished matrices are not similar matrices. We also give a
reasonable condition to fix the paramete
