_{1}(x)=x

^{0}+x

^{2}i +x

^{3}j +x

^{1}k

_{3}(x)=x

^{0}+x

^{2}i +x

^{1}j -x

^{3}k

^{0}+x

^{1}i +x

^{2}j +x

^{3}k

Similarly, the mapping

^{0}-x

^{1}i -x

^{2}j -x

^{3}k

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