quaternion space $\mathbb{H}_g$ is independent of the $S$-quaternion space $\mathbb{H}_e$ . The quaternion space $\mathbb{H}_g$ is suitable to describe the feature of gravitational field, while the $S$-quaternion space $\mathbb{H}_e$ is proper to depict the property of electromagnetic field.

In the quaternion space $\mathbb{H}_g$ for the gravitational field, the basis vector is $\mathbb{H}_g = ( \emph{\textbf{i}}_0, \emph{\textbf{i}}_1, \emph{\textbf{i}}_2, \emph{\textbf{i}}_3 )$, the radius vector is $\mathbb{R}_g = \emph{i} r_0 \emph{\textbf{i}}_0 + \Sigma r_k \emph{\textbf{i}}_k$, and the velocity is $\mathbb{V}_g = \emph{i} v_0 \emph{\textbf{i}}_0 + \Sigma v_k \emph{\textbf{i}}_k$. The gravitational potential is $\mathbb{A}_g = \emph{i} a_0 \emph{\textbf{i}}_0 + \Sigma a_k \emph{\textbf{i}}_k$, the gravitational strength is $\mathbb{F}_g = f_0 \emph{\textbf{i}}_0 + \Sigma f_k \emph{\textbf{i}}_k$, and the gravitational source is $\mathbb{S}_g = \emph{i} s_0 \emph{\textbf{i}}_0 + \Sigma s_k \emph{\textbf{i}}_k$. In the $\emph{S}$-quaternion space $\mathbb{H}_e$ for the electromagnetic field, the basis vector is $\mathbb{H}_e = ( \emph{\textbf{I}}_0, \emph{\textbf{I}}_1, \emph{\textbf{I}}_2, \emph{\textbf{I}}_3 )$, the radius vector is $\mathbb{R}_e = \emph{i} R_0 \emph{\textbf{I}}_0 + \Sigma R_k \emph{\textbf{I}}_k$, and the velocity is $\mathbb{V}_e = \emph{i} V_0 \emph{\textbf{I}}_0 + \Sigma V_k \emph{\textbf{I}}_k$. The electromagnetic potential is $\mathbb{A}_e = \emph{i} A_0 \emph{\textbf{I}}_0 + \Sigma A_k \emph{\textbf{I}}_k$, the electromagnetic strength is $\mathbb{F}_e = F_0 \emph{\textbf{I}}_0 + \Sigma F_k \emph{\textbf{I}}_k$, and the electromagnetic source 