4 dimensional numbers I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the product $ij$. So by adding another dimension, I've defined $$k=\begin{pmatrix}
1 & 0\\ 
0 & -1
\end{pmatrix}$$
with the property $k^2=1$. So numbers of the form $a+bi+cj+dk$ where ${{a,b,c,d}} \in \Bbb R^4$, $i$ is the imaginary unit, $j$ is the elementry unit of split complex numbers and k the number defined above, could be represented on a 4 dimensinal space. I know that these numbers look like the Quaternions. They are not! So far, I came out with the multiplication table below :
$$\begin{array}{|l |l l l|}\hline
  & i&j&k \\ \hline
i&-1&k&j \\ 
j& -k&1&i \\ 
k& -j&-i&1 \\ \hline
\end{array}$$
We can note that commutativity no longer exists with these numbers like the Quaternions. When I showed this work to my math teacher he said basicaly these :


*

*It's not coherent using numbers with different properties as basic element, since $i^2=-1$ whereas $j^2=k^2=1$

*2x2 matrices doesn't represent anything on a 4 dimensional space


Can somebody explains these 2 things to me. What's incoherent here?

Here is the Mathematica code for your proposed system, split-quaternions, using $2\times2$ real matrices $i=\left(
\begin{array}{cc}
 0 & 1 \\
 -1 & 0 \\
\end{array}
\right),j=\left(
\begin{array}{cc}
 0 & 1 \\
 1 & 0 \\
\end{array}
\right), k=\left(
\begin{array}{cc}
 1 & 0 \\
 0 & -1 \\
\end{array}
\right) $:
Unprotect[Dot];
Dot[x_?NumberQ, y_] := x y;
Protect[Dot];
Matrix /: Matrix[x_?MatrixQ] := 
  First[First[x]] /; x == First[First[x]] IdentityMatrix[Length[x]];
Matrix /: NonCommutativeMultiply[Matrix[x_?MatrixQ], y_] := 
  Dot[Matrix[x], y];
Matrix /: NonCommutativeMultiply[Matrix[y_, x_?MatrixQ]] := 
  Dot[y, Matrix[x]];
Matrix /: Dot[Matrix[x_], Matrix[y_]] := Matrix[x . y];
Matrix /: Matrix[x_] + Matrix[y_] := Matrix[x + y];
Matrix /: x_?NumericQ + Matrix[y_] := 
  Matrix[x IdentityMatrix[Length[y]] + y];
Matrix /: x_?NumericQ  Matrix[y_] := Matrix[x y];
Matrix /: Matrix[x_]*Matrix[y_] := Matrix[x . y] /; x . y == y . x;
Matrix /: Re[Matrix[x_?MatrixQ]] := Tr[x]/Length[x];
Matrix /: Conjugate[Matrix[x_?MatrixQ]] := Matrix[-x] + 2 Re[Matrix[x]]
Matrix /: Power[Matrix[x_ ?MatrixQ], y_] := 
  Matrix[MatrixPower[x, y]];

$Post = FullSimplify[# /. i -> Matrix[( {
                {0, 1},
                {-1, 0}
               } )] /. j -> Matrix[( {
               {0, 1},
               {1, 0}
              } )] /. k -> Matrix[( {
              {1, 0},
              {0, -1}
             } ) ] /. 
        f_[args1___, Matrix[mat_], args2___] :> 
         Matrix[MatrixFunction[f[args1, #, args2] &, mat]] /. 
       Matrix[{{a_, c_}, {d_, b_}}] :> (a + b)/
          2 + (c - d)/2 i + (c + d)/2 j + (a - b)/
           2 k ] &;

Test:
In=(i + k)^2

Out=0

In=Log[2 i + 3 j + 4]

Out=1/10 (2 Sqrt[5] (2 i + 3 j) ArcCoth[4/Sqrt[5]] + Log[161051])

In=(2 i + 3 j + 4) ** (5 - k)

Out=20 + 13 i + 17 j - 4 k

