@vv I have seldom used  odes in Maple so couldn't really attempt a solution. Just for visual effect I added these few lines to your code to see the paths in 3D.

plt0:=plots:-display(plottools:-tetrahedron(convert~([a0,b0,c0,d0],list)),color="White",transparency = 0.7):
plt1:=plots:-odeplot(sol, [aX(t),aY(t),aZ(t)], t=0..0.81, color=red):
plt2:=plots:-odeplot(sol, [bX(t),bY(t),bZ(t)], t=0..0.81, color=green):
plt3:=plots:-odeplot(sol, [cX(t),cY(t),cZ(t)], t=0..0.81, color=blue):
plt4:=plots:-odeplot(sol, [dX(t),dY(t),dZ(t)], t=0..0.81, color=purple):
plots:-display(plt0,plt1,plt2,plt3,plt4,scaling =constrained)

I asked a similar question a few years age. See Can a different Document or Worksheet template be created - MaplePrimes

@Kitonum Yes. That is basically  how I have being using gcd. But it is good to be reminded of the basics. 

This is part of a procedure I use for reducing vectors, lists, mactrices and expressions to remove common factors. Wondering now if frontend can be more efficiently combined into this. I hadn't thought about frontend when asking the quesdtion earlier on. 

proc(v::{list,Vector})
option overload;
         description " removes linear factor";
uses LinearAlgebra;
         local i, num,tgdc,dnm, V1;
        
            num:=`ifelse`(type(v,Vector),numelems(v),nops(v));
            dnm:=frontend(lcm, [seq(denom(v[i]),i=1..num)]);
            V1:=simplify(v*~dnm);
            tgdc:=V1[1];
            for i from 2 to num do
               tgdc:=frontend(gcd, [tgdc, V1[i]]);
            end do;
            return [tgdc/dnm, simplify(V1/~tgdc)];
end proc,

@Carl Love Thank you. Thak will make things so much simpler.

@acer Out of interest how did you discover/find the hidden character?

@mmcdara true/false is a valid answer for me. I was only solving for alpha as I couldn't think of any other way to derermine an asnwer.

@dharr  I see I tried it the wrong way around.

@acer  Thank you for pointing that out. I should try in future.

 Could you provide a sketch showing exactly what you mean. Say from Geogebra.

Is O the Origin?

Does it happen with rational numbers aswell

(123/100)/(n^(165/100)). 

I am not at my home pc to try.

@Kitonum Thank you. So it is sort of a little bug. I think I will keep the Tabulate and live with it.

@Kitonum Ok. But what I am looking for is needed in terms of varp. I can't get it to work.

varp:=alpha     but could be beta, lambda,  t etc.

`#3  Q24` (`&varp;`);  # not working

to give 

@acer Oh that is not a problem. You a busier than I and you find time to help a lot.

@acer That works perfectly. I would not have worked out the AddType for a long time. I did however manage to figure out the spellchecking by looking inside the Maple routine for Vector. But not as efficient as yours.
 

ProjVector := proc(a, b, c) 
      local cfs, vectr; 
      description " A Projective Line or Row Vector in Reduced format";
      if type(procname, 'indexed') then
         if procname::anyindex(identical(':-row')) then
            cfs := sign(c)*FactReduce([a, b, c]); 
            vectr := <[<cfs>^%T]>^%T;
            return vectr,type(vectr,ProjVR);
          elif procname::anyindex(identical(':-column')) then
            cfs := FactReduce([a, b, c]); 
            vectr := <[<cfs>]>;
            return vectr,type(vectr,ProjVC);
          else 
            error "either row or column is spelt incorrectly"; 
         end if;
       else
         cfs := FactReduce([a, b, c]); 
         vectr := <[<cfs>]>;
         return vectr,type(vectr,ProjVC);
      end if;
      end proc:

@acer I have just edited the question and put a link in.