Octonions from the Cayley-Dickson page. Table of $e_k = e_i \cdot e_j$.
$\begin{array}{c|cccccccc}
e_i \backslash e_j & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\
\hline
e_0 & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\
e_1 & e_1 & -e_0 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 \\
e_2 & e_2 & -e_3 & -e_0 & e_1 & e_6 & e_7 & -e_4 & -e_5 \\
e_3 & e_3 & e_2 & -e_1 & -e_0 & e_7 & -e_6 & e_5 & -e_4 \\
e_4 & e_4 & -e_5 & -e_6 & -e_7 & -e_0 & e_1 & e_2 & e_3 \\
e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -e_0 & -e_3 & e_2 \\
e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -e_0 & -e_1 \\
e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -e_0
\end{array}$
now to write this table in terms of vector operations:
$(e_{123} \vec{A}) (e_{123} \vec{B})$
$= (e_1 A_x + e_2 A_y + e_3 A_z)(e_1 B_x + e_2 B_y + e_3 B_z)$
$= e_1 A_x (e_1 B_x + e_2 B_y + e_3 B_z)
+ e_2 A_y (e_1 B_x + e_2 B_y + e_3 B_z)
+ e_3 A_z (e_1 B_x + e_2 B_y + e_3 B_z)$
$= e_1 e_1 A_x B_x + e_1 e_2 A_x B_y + e_1 e_3 A_x B_z
+ e_2 e_1 A_y B_x + e_2 e_2 A_y B_y + e_2 e_3 A_y B_z
+ e_3 e_1 A_z B_x + e_3 e_2 A_z B_y + e_3 e_3 A_z B_z$
$=-e_0 A_x B_x + e_3 A_x B_y - e_2 A_x B_z
- e_3 A_y B_x - e_0 A_y B_y + e_1 A_y B_z
+ e_2 A_z B_x - e_1 A_z B_y - e_0 A_z B_z$
$=-e_0 (A_x B_x + A_y B_y + A_z B_z)
+ e_1 (A_y B_z - A_z B_y)
+ e_2 (A_z B_x - A_x B_z)
+ e_3 (A_x B_y - A_y B_x)$
$= -e_0 \vec{A} \cdot \vec{B} + e_{123} \vec{A} \times \vec{B}$
$(e_{123} \vec{A}) (e_{567} \vec{B})$
$= (e_1 A_x + e_2 A_y + e_3 A_z)(e_5 B_x + e_6 B_y + e_7 B_z)$
$= e_1 A_x (e_5 B_x + e_6 B_y + e_7 B_z)
+ e_2 A_y (e_5 B_x + e_6 B_y + e_7 B_z)
+ e_3 A_z (e_5 B_x + e_6 B_y + e_7 B_z)$
$= e_1 e_5 A_x B_x + e_1 e_6 A_x B_y + e_1 e_7 A_x B_z
+ e_2 e_5 A_y B_x + e_2 e_6 A_y B_y + e_2 e_7 A_y B_z
+ e_3 e_5 A_z B_x + e_3 e_6 A_z B_y + e_3 e_7 A_z B_z$
$=-e_4 A_x B_x - e_7 A_x B_y + e_6 A_x B_z
+ e_7 A_y B_x - e_4 A_y B_y - e_5 A_y B_z
- e_6 A_z B_x + e_5 A_z B_y - e_4 A_z B_z$
$=-e_4 (A_x B_x + A_y B_y + A_z B_z)
- e_5 (A_y B_z - A_z B_y)
- e_6 (A_z B_x - A_x B_z)
- e_7 (A_x B_y - A_y B_x)$
$=-e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B}$
$(e_{567} \vec{A}) (e_{123} \vec{B})$
$= (e_5 A_x + e_6 A_y + e_7 A_z)(e_1 B_x + e_2 B_y + e_3 B_z)$
$= e_5 A_x (e_1 B_x + e_2 B_y + e_3 B_z)
+ e_6 A_y (e_1 B_x + e_2 B_y + e_3 B_z)
+ e_7 A_z (e_1 B_x + e_2 B_y + e_3 B_z)$
$= e_5 e_1 A_x B_x + e_5 e_2 A_x B_y + e_5 e_3 A_x B_z
+ e_6 e_1 A_y B_x + e_6 e_2 A_y B_y + e_6 e_3 A_y B_z
+ e_7 e_1 A_z B_x + e_7 e_2 A_z B_y + e_7 e_3 A_z B_z$
$= e_4 A_x B_x - e_7 A_x B_y + e_6 A_x B_z
+ e_7 A_y B_x + e_4 A_y B_y - e_5 A_y B_z
- e_6 A_z B_x + e_5 A_z B_y + e_4 A_z B_z$
$= e_4 (A_x B_x + A_y B_y + A_z B_z)
- e_5 (A_y B_z - A_z B_y)
- e_6 (A_z B_x - A_x B_z)
- e_7 (A_x B_y - A_y B_x)$
$= e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B}$
$(e_{567} \vec{A}) (e_{567} \vec{B})$
$= (e_5 A_x + e_6 A_y + e_7 A_z)(e_5 B_x + e_6 B_y + e_7 B_z)$
$= e_5 A_x (e_5 B_x + e_6 B_y + e_7 B_z)
+ e_6 A_y (e_5 B_x + e_6 B_y + e_7 B_z)
+ e_7 A_z (e_5 B_x + e_6 B_y + e_7 B_z)$
$= e_5 e_5 A_x B_x + e_5 e_6 A_x B_y + e_5 e_7 A_x B_z
+ e_6 e_5 A_y B_x + e_6 e_6 A_y B_y + e_6 e_7 A_y B_z
+ e_7 e_5 A_z B_x + e_7 e_6 A_z B_y + e_7 e_7 A_z B_z$
$=-e_0 A_x B_x - e_3 A_x B_y + e_2 A_x B_z
+ e_3 A_y B_x - e_0 A_y B_y - e_1 A_y B_z
- e_2 A_z B_x + e_1 A_z B_y - e_0 A_z B_z$
$=-e_0 (A_x B_x + A_y B_y + A_z B_z)
- e_1 (A_y B_z - A_z B_y)
- e_2 (A_z B_x - A_x B_z)
- e_3 (A_x B_y - A_y B_x)$
$= -e_0 \vec{A} \cdot \vec{B} - e_{123} \vec{A} \times \vec{B}$
In shorthand:
$\begin{array}{c|cccc}
& e_0 b & e_{123} \vec{B} & e_4 b & e_{567} \vec{B} \\
\hline
e_0 a & e_0 a b & e_{123} a \vec{B} & e_4 a b & e_{567} a \vec{B} \\
e_{123} \vec{A} & e_{123} \vec{A} b & -e_0 \vec{A} \cdot \vec{B} + e_{123} \vec{A} \times \vec{B} & e_{567} \vec{A} b & -e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B} \\
e_4 a & e_4 a b & -e_{567} a \vec{B} & -e_0 a b & e_{123} a \vec{B} \\
e_{567} \vec{A} & e_{567} \vec{A} b & e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B} & -e_{123} \vec{A} b & -e_0 \vec{A} \cdot \vec{B} - e_{123} \vec{A} \times \vec{B}
\end{array}$
This is just an idea of incorporating vector notation. Not a good one though. What would be a better one?
$J = e_1 J_x + e_2 J_y + e_3 J_z = e_{123} \vec{J}$
$E = e_1 E_x + e_2 E_y + e_3 E_z = e_{123} \vec{E}$
$B = e_1 B_x + e_2 B_y + e_3 B_z = e_{123} \vec{B}$
$F = E - e_4 B$
$F = e_1 E_x + e_2 E_y + e_3 E_z + e_5 B_x + e_6 B_y + e_7 B_z$
$F = e_{123} \vec{E} + e_{567} \vec{B}$
$D = e_1 \partial_x + e_2 \partial_y + e_3 \partial_z - e_4 \partial_t = e_{123} \vec{\nabla} - e_4 \partial_t$
$K = e_4 (J + e_4 \rho)$
$K = -e_0 \rho - e_5 J_x - e_6 J_y - e_7 J_z$
$K = -e_0 \rho - e_{567} \vec{J}$
$D F = K$
...gives...
$(e_1 \partial_x + e_2 \partial_y + e_3 \partial_z - e_4 \partial_t) (e_1 E_x + e_2 E_y + e_3 E_z + e_5 B_x + e_6 B_y + e_7 B_z) = -e_0 \rho - e_5 J_x - e_6 J_y - e_7 J_z$
$(e_{123} \vec{\nabla} - e_4 \partial_t) ( e_{123} \vec{E} + e_{567} \vec{B}) = -e_0 \rho - e_{567} \vec{J}$
$e_{123} \vec{\nabla} e_{123} \vec{E}
- e_4 \partial_t e_{123} \vec{E}
+ e_{123} \vec{\nabla} e_{567} \vec{B}
- e_4 \partial_t e_{567} \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
$-e_0 (\vec{\nabla} \cdot \vec{E}) + e_{123} (\vec{\nabla} \times \vec{E})
+ e_{567} \partial_t \vec{E}
- e_4 \vec{\nabla} \cdot \vec{B} - e_{567} (\vec{\nabla} \times \vec{B})
- e_{123} \partial_t \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
unraveled:
$\begin{matrix}
e_0: & \vec{\nabla} \cdot \vec{E} = \rho \\
e_4: & \vec{\nabla} \cdot \vec{B} = 0 \\
e_{123}:& -\partial_t \vec{B} + \vec{\nabla} \times \vec{E} = 0 \\
e_{567}:& -\partial_t \vec{E} + \vec{\nabla} \times \vec{B} = \vec{J}
\end{matrix}$
alternative signs:
$(e_{123} \vec{\nabla} + e_4 \partial_t) ( e_{123} \vec{E} + e_{567} \vec{B}) = -e_0 \rho - e_{567} \vec{J}$
$
e_{123} \vec{\nabla} e_{123} \vec{E}
+ e_4 \partial_t e_{123} \vec{E}
+ e_{123} \vec{\nabla} e_{567} \vec{B}
+ e_4 \partial_t e_{567} \vec{B}
= -e_0 \rho - e_{567} \vec{J}
$
$
-e_0 \vec{\nabla} \cdot \vec{E}
+ e_{123} \vec{\nabla} \times \vec{E}
- e_{567} \partial_t \vec{E}
- e_4 \vec{\nabla} \cdot \vec{B}
- e_{567} \vec{\nabla} \times \vec{B}
+ e_{123} \partial_t \vec{B}
= -e_0 \rho - e_{567} \vec{J}
$
$
\vec{\nabla} \cdot \vec{E} = \rho;
\vec{\nabla} \cdot \vec{B} = 0;
\partial_t \vec{B} + \vec{\nabla} \times \vec{E} = 0;
- \partial_t \vec{E} - \vec{\nabla} \times \vec{B} = - \vec{J}
$
$(e_{123} \vec{\nabla} + e_4 \partial_t) ( e_{123} \vec{E} - e_{567} \vec{B}) = -e_0 \rho - e_{567} \vec{J}$
$
e_{123} \vec{\nabla} e_{123} \vec{E}
+ e_4 \partial_t e_{123} \vec{E}
- e_{123} \vec{\nabla} e_{567} \vec{B}
- e_4 \partial_t e_{567} \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
$
-e_0 \vec{\nabla} \cdot \vec{E}
+ e_{123} \vec{\nabla} \times \vec{E}
- e_{567} \partial_t \vec{E}
+ e_4 \vec{\nabla} \cdot \vec{B}
+ e_{567} \vec{\nabla} \times \vec{B}
- e_{123} \partial_t \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
$
\vec{\nabla} \cdot \vec{E} = \rho;
\vec{\nabla} \cdot \vec{B} = 0;
-\partial_t \vec{B} + \vec{\nabla} \times \vec{E} = 0;
-\partial_t \vec{E} + \vec{\nabla} \times \vec{B} = -\vec{J}$
$(e_{123} \vec{\nabla} - e_4 \partial_t) ( e_{123} \vec{E} + e_{567} \vec{B}) = -e_0 \rho - e_{567} \vec{J}$
$
e_{123} \vec{\nabla} e_{123} \vec{E}
- e_4 \partial_t e_{123} \vec{E}
+ e_{123} \vec{\nabla} e_{567} \vec{B}
- e_4 \partial_t e_{567} \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
$
-e_0 \vec{\nabla} \cdot \vec{E}
+ e_{123} \vec{\nabla} \times \vec{E}
+ \partial_t e_{567} \vec{E}
- e_4 \vec{\nabla} \cdot \vec{B}
- e_{567} \vec{\nabla} \times \vec{B}
- e_{123} \partial_t \vec{B}
= -e_0 \rho - e_{567} \vec{J}$
$
\vec{\nabla} \cdot \vec{E} = \rho;
\vec{\nabla} \cdot \vec{B} = 0;
- \partial_t \vec{B} + \vec{\nabla} \times \vec{E} = 0;
- \partial_t \vec{E} + \vec{\nabla} \times \vec{B} = \vec{J}$
$(e_{123} \vec{\nabla} - e_4 \partial_t) ( e_{123} \vec{E} - e_{567} \vec{B}) = -e_0 \rho - e_{567} \vec{J}$
Desired:
$\vec{\nabla} \cdot \vec{E} = \rho$
$\vec{\nabla} \cdot \vec{B} = 0$
$\partial_t B + \vec{\nabla} \times \vec{E} = 0$
$-\partial_t E + \vec{\nabla} \times \vec{B} = \vec{J}$
So there's one sign off. How to fix it. Turns out you can't do $D F = K$ using the $e_i$ basis. You can, however, do it using anti-associativity of certain octonion basis elements:
$D E - (D e_4) B = K$
gives us our desired results.