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.

The algebra you discribed in the question is tessarines.
So, there are two matrical representations of tessarines.

*

*For a tessarine $z=w_1+w_2i+w_3j+w_4ij$ the matrix representation is as follows:

$$\left(
\begin{array}{cccc}
 {w_0} & -{w_1} & {w_2} & -{w_3} \\
 {w_1} & {w_0} & {w_3} & {w_2} \\
 {w_2} & -{w_3} & {w_0} & -{w_1} \\
 {w_3} & {w_2} & {w_1} & {w_0} \\
\end{array}
\right)$$
One has to use the 4x4 matrix. After performing the operations, the coefficients of the resulting tessarine can be extracted from the first column of the resulting matrix.


*Alternatively, one can use a 2x2 matrix of the form $z=a+bj=\left(
\begin{array}{cc}
 a & b \\
 b & a \\
\end{array}
\right)$
where both $a$ and $b$ are complex numbers. Since complex numbers are embedded in most computer algebra systems, I prefer the second method.
