Linear mapping of quaternion algebra

Linear automorphism of quaternion algebra is a liner over real field mapping $$f:H\rightarrow H$$ such that $$f(ab)=f(a)f(b)$$ Evidently, linear mapping $$E(x)=x$$ is linear automorphism. In quaternion algebra there are nontrivial linear automorphisms. For instance
E₁(x)=x⁰ +x²i +x³j +x¹k E₃(x)=x⁰ +x²i +x¹j -x³k where
x=x⁰ +x¹i +x²j +x³k

Similarly, the mapping

I(x)=x⁰ -x¹i -x²j -x³k is antilinear automorphism because $$I(ab)=I(b)I(a)$$

In the paper eprint arXiv:1107.1139, Linear Mappings of Quaternion Algebra, I proved following statement. For any linear over real field function $f$ there is unique expansion

$$f(x)=a_0E(x)+a_1E_1(x)+a_2E_2(x)+a_3I(x)$$