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.