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?

You have discovered split-quaternions. You can compare the multiplication table there and in your question.
This algebra is not commutative and has zero divisors. So, it combines the "negative" traits of both quaternions and tessarines. On the other hand it is notable to be isimorphic to the $2\times2$ matrices. Due to this isomorphism, people usually speak about matrices rather than split-quaternions.
