"Optimize 3-fold nested loop given set of comparision-conditions between loop-variables" or "optimize lower-/upper bounds"

Question

For a mathematical problem I want to find solutions which involve 3 loop-variables, say (e,h,i). I have a set of lower/greater-conditions on the loop-variables, so there is a real chance to reduce the search-space from -say [1..100,1..100,1..100] radically, if -say- e is given.
The basic condition is, that in fact I work with a set of 6 linear combinations of e²,h²,i² taken as polynomials which should as well result in a square value, thus a positive value. For instance the first condition is 2e² - i² = a² and the second is 2e²-h²=b²

----------------------
  *ee *hh *ii
----------------------
   2   0  -1  = aa
   2  -1   0  = bb
  -1   1   1  = cc
  -2   1   2  = dd
   1   0   0  = ee        // shown just for completeness
   4  -1  -2  = ff
   3  -1  -1  = gg
   0   1   0  = hh        // shown just for completeness
   0   0   1  = ii        // shown just for completeness

I write ee,hh,ii for ease of notation. From the first condition, to get a positive value a² I must thus have 2ee > ii so for a value in loop-variable e I know the upper bound for the loop-variable i is an integer near sqrt(2ee).

Here are the 6 relevant conditions:

   2ee  >       ii
   2ee  >   hh
   -ee  >  -hh- ii
  -2ee  >  -hh-2ii
   4ee  >   hh+2ii
   3ee  >   hh+ ii

The loop-construct shall finally be like this and the upper bound for e as large as possible to run in acceptable time (this is actually pseudocode!)

  for e=1 to 1000; ee=e^2 ; hlb=<some-lower-bound>; hub=<some-upper-bound>;
     for h=hlb to hub; hh=h^2 ; ilb=<some-lower-bound>; iub=<some-upper-bound>;
        for i=ilb to iub; ii=i^2;
               aa=2*ee-ii; if !issquare(aa) then next;
               bb=2*ee-hh; if !issquare(bb) then next;
               cc= -ee+hh+ii; if !issquare(cc) then next;
               <and so on; I expext at least 4 squares in aa,bb,cc,dd,ff,gg>
               if condition-is-met then print("one solution:",e,h,i);
        next i;
     next h;
   next e;

My questions is: what are the tightest bounds for the loop-variables h and i given e. I'd most like to have a general ansatz to solve such a problem.

What I've found so far by combination and aggregation of the conditions

           2ee  >    ii,hh
 hh+ ii >   ee    
 hh+2ii >  2ee  
           4ee  >   hh+2ii
           3ee  >   hh+ ii

then

             12ee  >   6ii,6hh
12hh+12ii >  12ee    
 6hh+12ii >  12ee  
             12ee  >   3hh+6ii
             12ee  >   4hh+4ii

using the smallest upper bounds (on lhs) resp largest lower bounds (on rhs) I think I can simplify

24ee > 6hh+12ii >  12ee  >   6hh,6ii, 4hh+4ii

But then I do not see how I could simplify this more and actually formulate hlb,hub,ilb,iub ...