I have the following code, which allows evaluating expressions with split-complex unity J:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
$Pre = If[FreeQ[#, J], #, Module[{tmp},
          
     tmp = Evaluate[
        MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] // 
       FullSimplify;
           tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

After first run it allows evaluating functions of J just as if it was an embedded constant. But trying to slove equations with it fails, and returns useless output. Can this code be modified so that Solve and Reduce also would work?

An example of possible equations:

Solve[(a+b J)^2==J,{a,b}]  Solve[(a+b J)^(a+b J)==1,{a,b}]

I have the following code, which allows evaluating expressions with split-complex unity J:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
$Pre = If[FreeQ[#, J], #, Module[{tmp},
          
     tmp = Evaluate[
        MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] // 
       FullSimplify;
           tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

After first run it allows evaluating functions of J just as if it was an embedded constant. But trying to slove equations with it fails, and returns useless output. Can this code be modified so that Solve and Reduce also would work?

An example of possible equations:

Solve[(a+b J)^2==1,{a,b}]

Solve[(a+b J)^2 == (a + b J), {a, b}]

Solve[(a+b J)^2==J,{a,b}] 

Solve[(a+b J)^(a+b J)==1,{a,b}]

I have the following code, which allows evaluating expressions with split-complex unity J:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
$Pre = If[FreeQ[#, J], #, Module[{tmp},
          
     tmp = Evaluate[
        MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] // 
       FullSimplify;
           tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

After first run it allows evaluating functions of J just as if it was an embedded constant. But trying to slove equations with it fails, and returns useless output. Can this code be modified so that Solve and Reduce also would work?

I have the following code, which allows evaluating expressions with split-complex unity J:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
$Pre = If[FreeQ[#, J], #, Module[{tmp},
          
     tmp = Evaluate[
        MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] // 
       FullSimplify;
           tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

After first run it allows evaluating functions of J just as if it was an embedded constant. But trying to slove equations with it fails, and returns useless output. Can this code be modified so that Solve and Reduce also would work?

An example of possible equations:

Solve[(a+b J)^2==J,{a,b}] Solve[(a+b J)^(a+b J)==1,{a,b}]