How to implement split-complex numbers?

Question

For those who do not know, the split-complex numbers are an analogue to the complex numbers where J is defined such that $J^2=1$ but $Jnepm1$, so they are all of the form $a+bJ$.

By using TagSetDelayed, I tried to define the split-complex numbers as so:

J /: J^2 := 1

If I then type J^2, I get the output 1.
However, if I type J^3, I just get the output J^3. I would like to instead get the output $J$, since $J^3=J^2J=1J=J$.
Is there a better way to implement this number system?

Henrik Schumacher · Accepted Answer

Try this:

J /: Power[J, p_Integer?OddQ] := J
J /: Power[J, p_Integer?EvenQ] := 1

J^Range[-10, 10]

{1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1}

swish · Answer

You can represent split-complex numbers as multivectors (geometric number, clifford number) with signature $(1, 0)$. I have a paclet for this:
PacletInstall["https://wolfr.am/OkONsyY2"]

<< GeometricAlgebra`

(* this is your split-complex number *)
h = Multivector[{a, b}, 1]

(* you can perform numeric operations with it *)
h^2
h^-1
MultivectorFunction[Exp, h] (* similar to MatrixFunction *)

How to implement split-complex numbers?

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