Octonions from the Cayley-Dickson page:
$\matrix{ & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_0 & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_1 & e_1 & -e_0 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 \\ e_2 & e_2 & -e_3 & -e_0 & e_1 & e_6 & e_7 & -e_4 & -e_5 \\ e_3 & e_3 & e_2 & -e_1 & -e_0 & e_7 & -e_6 & e_5 & -e_4 \\ e_4 & e_4 & -e_5 & -e_6 & -e_7 & -e_0 & e_1 & e_2 & e_3 \\ e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -e_0 & -e_3 & e_2 \\ e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -e_0 & -e_1 \\ e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -e_0 \\ }$

in shorthand?:
$\matrix{ & e_0 a & e_{123} \vec{A} & e_4 a & e_{567} \vec{A} \\ e_0 b & e_0 a b & e_{123} \vec{A} b & e_4 a & e_{567} \vec{A} b \\ e_{123} \vec{B} & e_{123} a \vec{B} & -e_0 \vec{A} \cdot \vec{B} + e_{123} \vec{A} \times \vec{B} & e_{567} a \vec{B} & -e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B} \\ e_4 b & e_4 a b & -e_{567} \vec{A} b & -e_0 a b & e_{123} \vec{A} b \\ e_{567} \vec{B} & e_{567} a \vec{B} & e_4 \vec{A} \cdot \vec{B} - e_{567} \vec{A} \times \vec{B} & -e_{123} a \vec{B} & -e_0 \vec{A} \cdot \vec{B} - e_{123} \vec{A} \times \vec{B} }$

This is just an idea of incorporating vector notation. Not a good one though. What would be a better one?

---

$J = e_1 J_x + e_2 J_y + e_3 J_z$

$E = e_1 E_x + e_2 E_y + e_3 E_z$

$B = e_1 B_x + e_2 B_y + e_3 B_z$

$F = E - e_4 B$
$F = e_1 E_x + e_2 E_y + e_3 E_z + e_5 B_x + e_6 B_y + e_7 B_z$

$D = e_1 \partial_x + e_2 \partial_y + e_3 \partial_z - e_4 \partial_t$

$K = e_4 (J + e_4 \rho)$
$K = -e_0 \rho - e_5 J_x - e_6 J_y - e_7 J_z$

$D F = K$

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:
$\begin{matrix} e_0: & \vec{\nabla} \cdot \vec{E} = \rho \\ e_4: & \vec{\nabla} \cdot \vec{B} = 0 \\ e_{123}:& \partial_t \vec{B} + \vec{\nabla} \times \vec{E} = 0 \\ e_{567}:& -\partial_t \vec{E} + \vec{\nabla} \times \vec{B} = \vec{J} \end{matrix}$