Is split-complex $j=i+2\epsilon$? In matrix representation 
imaginary unit 
$$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$
dual numbers unit
$$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$
split-complex unit
$$j=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$
Given this definition, does not it follow that
$$j=i+2\epsilon$$
and as such, one of these systems can be fully expressed through others?

In the hindsight this question looks a bit stupid to me, but now I just want to answer it myself.
The set of the $2\times2$ matrices is isomorphic to the set of split-quaternions.
That said, in split-quaternions, $i$ is usually defined as
$$i=\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$$
(which has the opposite sign from the question, but the both definitions are equal up to isomorphism).
$$j=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$
(as in the question).
The nilpotent element $\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$ indeed generates an algebra, which is isomorphic to dual numbers.
Thus, the proposed equalities are correct, but one should bear in mind that the algebra of split-quaternions is not commutative. If one would want a commutative algebra with elements of similar properties, one possibly should consider something like dual tessarines, which is $8$-dimensional.
