is $\mathbb{S}_e = i S_0 \emph{\textbf{I}}_0 + \Sigma S_k \emph{\textbf{I}}_k$. Herein $\mathbb{H}_e = \mathbb{H}_g \circ \emph{\textbf{I}}_0$. The symbol $\circ$ denotes the octonion multiplication. $r_j, ~v_j, ~a_j, ~s_j, ~R_j, ~V_j, ~A_j, ~S_j, ~f_0$, and $F_0$ are all real. $f_k$ and $F_k$ are the complex numbers. $i$ is the imaginary unit. $\emph{\textbf{i}}_0 = 1$. $j = 0, ~1, ~2, ~3$. $k = 1, ~2, ~3$.

These two orthogonal quaternion spaces, $\mathbb{H}_g$ and $\mathbb{H}_e$, composes one octonion space, $\mathbb{O} = \mathbb{H}_g + \mathbb{H}_e$. In the octonion space $\mathbb{O}$ for the electromagnetic and gravitational fields, the octonion radius vector is $\mathbb{R} = \mathbb{R}_g + k_{eg} \mathbb{R}_e$, the octonion velocity is $\mathbb{V} = \mathbb{V}_g + k_{eg} \mathbb{V}_e$ , with $k_{eg}$ being one coefficient. Meanwhile the octonion field potential is $\mathbb{A} = \mathbb{A}_g + k_{eg} \mathbb{A}_e$, the octonion field strength is $\mathbb{F} = \mathbb{F}_g + k_{eg} \mathbb{F}_e$. From here on out, some physics quantities are extended from the quaternion functions to the octonion functions, according to the characteristics of octonion. Apparently $\mathbb{V}$, $\mathbb{A}$, and their differential coefficients are all octonion functions of $\mathbb{R}$.

The octonion definition of field strength is, $$\mathbb{F} = \square \circ \mathbb{A}  ~,$$ where $\mathbb{F}_g = \square \circ \mathbb{A}_g$, $\mathbb{F}_e = \square \circ 