Understanding bicomplex numbers I found by chance, the set of Bicomplex numbers. These numbers took particularly my attention because of their similarity to my previous personal research and question. I should say that I can't really understand the fact that $j^2=+1$ (and must of other abstract algebra) without using matricial interpretations. When I look a bit on the Bicomplex numbers, the think that surprised me a lot was the fact that $ij=ji=k$ and $k^2=-1$. Because using matricial representations, we get :
$$ij=\begin{pmatrix}
0 & -1\\ 
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & 1\\ 
1 & 0
\end{pmatrix}=\begin{pmatrix}
-1 & 0\\ 
0 & 1
\end{pmatrix}$$
$$ji=\begin{pmatrix}
0 & 1\\ 
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & -1\\ 
1 & 0
\end{pmatrix}=\begin{pmatrix}
1 & 0\\ 
0 & -1
\end{pmatrix}$$
So, $ij$ should be different then $ji$. Then if we define $k=ij$ or $k=ji$, we get both way $$k^2= I$$
Looked at the Wikipedia article and did some internet search but couldn't find a matricial representation for the bicomplex $k$. So I wanna ask some clarifications about where from the relations $ij=ji$ and $k^2=-1$ come from.

In Mathematica, the following code assd support for the split-complex unity J along with complex unity I. This allows to calculate any expressions with tessarines:
 $Pre = If[FreeQ[#, J], #, Module[{tmp},
      
 tmp = Evaluate[
    MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] // 
   FullSimplify;
       tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

J^I

Out= 1/2 (1 + (-1)^I) + 1/2 J (1 - Cosh[Pi] + Sinh[Pi])

