@Scot Gould Thank you. I will experiment with that. 

@Christopher2222 This is just a guess. Possibly the recent files only gets updated when one shuts down Maple. A sudden power failure, would not have time to do that. I know from using Solidworks, the settings etc are not saved until the one exits and restarts the program. So if one has changed some settings and then a crash they are lost. 

I tried  eliminating pairs [x,y] ,[ x,z] and [y,z]. That doesnt give a consistant answer though but maybe it will shed some light on the issue.

Download 2024-21-31_eliminate_test.mw

@acer  Yes SetMetric is a good way of avoiding accidently changing the Metric. Why use Metric:=copy(M) instead of Metric:=M?

@nm I was completly unaware of option object. This is a level of coding I have never come across. I know I didn't state this  or put it in the example given. I am using option package which it turns out can't be used with object.  When I put my example together, I thought, "I don't need that for this question".

I need to have a default value for the Metric matrix when the package is loaded. Can the object concept be used  inside a package? Is there any other way in this case? 

Globals may be bad, but I have used quite a few of them. I admit my thinking may be limited.

@Joe Riel  I like that. Both comments (now answers ) are helpful to know.

@Joe Riel Thanks. Had tried with no forward quotes at all previously. I find it rather subtle when to use and not to use them.

@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.