Map of Conjugation of Quaternion algebra

Quaternion algebra has following maps of conjugation
\begin{split}
x=E\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i+x_2j+x_3k\\
x^{*_1}=I\circ(x_0+x_1i+x_2j+x_3k)&=x_0-x_1i+x_2j+x_3k\\
x^{*_2}=J\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i-x_2j+x_3k\\
x^{*_3}=K\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i+x_2j-x_3k
\end{split}

A linear map of quaternion algebra
\[f:H\rightarrow H\ \ \ y^i=f^i_jx^j\]
has form
\[
f=a_0\circ E+a_1\circ I+a_2\circ J+a_3\circ K
\]
\begin{split}
f\circ x&=a_0x+a_1\circ I\circ x+a_2\circ J\circ x+a_3\circ K\circ x\\
&=a_0x+a_1x^{*_1}+a_2x^{*_2}+a_3x^{*_3}
\end{split}
where quaternions \(a_0\), \(a_1\), \(a_2\), \(a_3\) are defined by coordinates \(f^i_j\) of linear map.

The set \(\mathcal L(R;H;H)\) of linear maps of quaternion algebra is left \(H\)-vector space and has the basis \(\overline{\overline{e}} =(E,I,J,K)\).

The set \[HE=\{a\circ E:a\in H\}\] is \(R\)-algebra isomorphic to quaternion algebra.