I was looking for integer solutions to this equation: $$a^2+b^2+c^2=6d^2$$ And found a parametric solution. Given u, v, w : $$a=(u-v)^2-2(u+w)^2+3w^2$$ $$b=(u-v)^2-2(v+w)^2+3w^2$$ $$c=(u-v)^2+(u+v-w)^2-3w^2$$ $$d=u^2+v^2+w^2$$
When will a parametric solution generate all possible solutions?
a^2 + b^2 + c^2 /. {a -> (u - v)^2 - 2 (u + w)^2 + 3 w^2,
b -> (u - v)^2 - 2 (v + w)^2 + 3 w^2,
c -> (u - v)^2 + (u + v - w)^2 - 3 w^2} // Factor
Mathematica Code
On
I wrote a more careful program. It reports just one occurrence of each quadruple by the wxyz formulas i the first answer. I combined that with a raw search, so the computer would scream at me if any quadruple were missed. Success all around...
$$\color{magenta}{ a = ( w+x+z)^2 + ( -w + x + y)^2 - 3 y^2 - 3 z^2 } $$ $$\color{magenta}{ b = ( w + x - 2 z)^2 - 2 ( x-y-z)^2 + 3 ( y+z)^2 - 6 z^2 } $$ $$\color{magenta}{ c = ( w - x + 2 y)^2 - 2 ( x-y-z)^2 - 3 ( y-z)^2 + 6 z^2 } $$ $$\color{magenta}{ d = w^2 + x^2 + y^2 + z^2 } $$
1 a,b,c 2 1 1
1 a,b,c 2 -1 -1 w,x,y,z 0 -1 0 0
3 a,b,c -2 1 -7 w,x,y,z 0 -1 -1 1
3 a,b,c 2 5 5 w,x,y,z 0 -1 -1 -1
3 a,b,c 5 5 2
3 a,b,c 7 2 1
5 a,b,c 10 7 1
5 a,b,c -10 7 1 w,x,y,z 0 0 -2 -1
5 a,b,c 11 5 2
5 a,b,c 2 -5 -11 w,x,y,z 0 -2 0 1
7 a,b,c 10 -5 -13 w,x,y,z 1 -2 -1 1
7 a,b,c 13 10 5
7 a,b,c 13 11 2
7 a,b,c 17 2 1
7 a,b,c 2 -13 11 w,x,y,z 1 -2 1 -1
7 a,b,c -2 -17 -1 w,x,y,z 1 -2 1 1
9 a,b,c 10 19 5 w,x,y,z 0 -2 -2 -1
9 a,b,c -14 -11 -13 w,x,y,z 0 -2 1 2
9 a,b,c -14 -1 -17 w,x,y,z 0 -1 -2 2
9 a,b,c 14 13 11
9 a,b,c 17 14 1
9 a,b,c 19 10 5
9 a,b,c 19 11 2
9 a,b,c 2 11 -19 w,x,y,z 0 -2 -2 1
9 a,b,c 22 1 1
9 a,b,c -22 -1 -1 w,x,y,z 0 -1 2 2
11 a,b,c -10 25 1 w,x,y,z 0 -1 -3 -1
11 a,b,c 14 1 -23 w,x,y,z 0 -3 -1 1
11 a,b,c -14 13 -19 w,x,y,z 0 -1 -3 1
11 a,b,c 19 14 13
11 a,b,c 19 19 2
11 a,b,c 2 -19 -19 w,x,y,z 0 -3 1 1
11 a,b,c 23 14 1
11 a,b,c 25 10 1
11 a,b,c 26 5 5
11 a,b,c 26 5 5 w,x,y,z 0 -3 -1 -1
11 a,b,c 26 7 1
11 a,b,c -26 -7 1 w,x,y,z 0 -1 1 3
13 a,b,c 10 25 17 w,x,y,z 1 -2 -2 -2
13 a,b,c -14 -23 17 w,x,y,z 1 -2 2 -2
13 a,b,c -2 -13 -29 w,x,y,z 0 -3 0 2
13 a,b,c 22 19 13
13 a,b,c 22 19 -13 w,x,y,z 0 -3 -2 0
13 a,b,c -22 -23 1 w,x,y,z 1 -2 2 2
13 a,b,c 23 17 14
13 a,b,c 23 22 1
13 a,b,c 25 17 10
13 a,b,c 26 17 7
13 a,b,c -26 17 7 w,x,y,z 0 0 -3 -2
13 a,b,c 2 -7 -31 w,x,y,z 1 -2 -2 2
13 a,b,c 29 13 2
13 a,b,c 31 7 2
15 a,b,c 10 -17 -31 w,x,y,z 1 -3 -1 2
15 a,b,c -14 25 23 w,x,y,z 1 -1 -2 -3
15 a,b,c 22 5 -29 w,x,y,z 1 -3 -2 1
15 a,b,c -2 -35 11 w,x,y,z 1 -3 2 -1
15 a,b,c 25 23 14
15 a,b,c 26 25 7
15 a,b,c 26 7 25 w,x,y,z 1 -3 -1 -2
15 a,b,c 29 22 5
15 a,b,c 31 17 10
15 a,b,c 34 13 5
15 a,b,c -34 -13 5 w,x,y,z 1 -1 3 -2
15 a,b,c 35 11 2
17 a,b,c -14 13 -37 w,x,y,z 0 -2 -3 2
17 a,b,c 22 -17 -31 w,x,y,z 0 -4 0 1
17 a,b,c -22 -25 -25 w,x,y,z 0 -3 2 2
17 a,b,c -22 -5 -35 w,x,y,z 0 -2 -2 3
17 a,b,c 2 37 19 w,x,y,z 0 -2 -3 -2
17 a,b,c 25 25 22
17 a,b,c 26 23 23
17 a,b,c 26 23 23 w,x,y,z 0 -3 -2 -2
17 a,b,c 2 7 -41 w,x,y,z 0 -3 -2 2
17 a,b,c 31 22 17
17 a,b,c 34 23 7
17 a,b,c -34 23 -7 w,x,y,z 0 0 -4 -1
17 a,b,c 35 22 5
17 a,b,c 37 14 13
17 a,b,c 37 19 2
17 a,b,c 38 1 -17 w,x,y,z 0 -4 -1 0
17 a,b,c 38 13 11
17 a,b,c -38 -13 -11 w,x,y,z 0 -2 2 3
17 a,b,c 38 17 1
17 a,b,c 41 7 2
19 a,b,c 10 29 -35 w,x,y,z 0 -3 -3 1
19 a,b,c -14 -11 -43 w,x,y,z 0 -3 -1 3
19 a,b,c 14 -41 -17 w,x,y,z 1 -4 1 1
19 a,b,c -22 29 29 w,x,y,z 0 -1 -3 -3
19 a,b,c 22 41 1 w,x,y,z 0 -3 -3 -1
19 a,b,c -26 -23 -31 w,x,y,z 0 -3 1 3
19 a,b,c 26 -37 11 w,x,y,z 1 -4 1 -1
19 a,b,c 29 29 22
19 a,b,c 31 26 23
19 a,b,c 34 -13 -29 w,x,y,z 1 -4 -1 1
19 a,b,c 34 29 13
19 a,b,c 34 31 7
19 a,b,c -34 -7 -31 w,x,y,z 0 -1 -3 3
19 a,b,c 35 29 10
19 a,b,c 37 26 11
19 a,b,c 41 17 14
19 a,b,c 41 22 1
19 a,b,c 43 14 11
19 a,b,c 46 -1 7 w,x,y,z 1 -4 -1 -1
19 a,b,c 46 5 5
19 a,b,c -46 5 5 w,x,y,z 0 -1 3 3
19 a,b,c 46 7 1
21 a,b,c -2 -31 -41 w,x,y,z 0 -4 1 2
21 a,b,c 26 -17 -41 w,x,y,z 2 -3 -2 2
21 a,b,c -26 43 11 w,x,y,z 0 -1 -4 -2
21 a,b,c -34 11 -37 w,x,y,z 0 -1 -4 2
21 a,b,c 34 31 23
21 a,b,c 34 31 23 w,x,y,z 2 -3 -2 -2
21 a,b,c 37 34 11
21 a,b,c -38 -19 -29 w,x,y,z 0 -2 -1 4
21 a,b,c 38 29 19
21 a,b,c 41 26 17
21 a,b,c 41 31 2
21 a,b,c 43 26 11
21 a,b,c 46 19 13
21 a,b,c -46 -19 -13 w,x,y,z 0 -2 1 4
21 a,b,c 46 23 1
21 a,b,c 46 23 1 w,x,y,z 0 -4 -2 -1
21 a,b,c 50 11 5
21 a,b,c -50 -5 11 w,x,y,z 0 -1 2 4
23 a,b,c -10 -43 35 w,x,y,z 1 -3 2 -3
23 a,b,c 10 7 -55 w,x,y,z 1 -3 -3 2
23 a,b,c -14 -13 -53 w,x,y,z 1 -2 -3 3
23 a,b,c -2 -19 -53 w,x,y,z 1 -3 -2 3
23 a,b,c 22 29 43 w,x,y,z 1 -3 -2 -3
23 a,b,c -22 -49 17 w,x,y,z 1 -3 3 -2
23 a,b,c -2 47 31 w,x,y,z 1 -2 -3 -3
23 a,b,c 26 47 17 w,x,y,z 1 -3 -3 -2
23 a,b,c -34 -43 -13 w,x,y,z 1 -3 2 3
23 a,b,c 38 37 19
23 a,b,c -38 -37 19 w,x,y,z 1 -2 3 -3
23 a,b,c -38 -41 -7 w,x,y,z 1 -3 3 2
23 a,b,c 41 38 7
23 a,b,c 43 29 22
23 a,b,c 43 34 13
23 a,b,c 43 35 10
23 a,b,c 47 26 17
23 a,b,c 47 31 2
23 a,b,c 49 22 17
23 a,b,c 50 25 7
23 a,b,c -50 -25 7 w,x,y,z 1 -2 3 3
23 a,b,c 53 14 13
23 a,b,c 53 19 2
23 a,b,c 55 10 7
The quick way is to take quaternions with integer coefficients, say $$ q = w + x i + y j + z k, $$ with $$ i^2 = j^2 = k^2 = -1, $$ then $$ ij = k, jk = i, k i = j $$ in cyclic order, then $$ ji = -k, kj = -i, ik = -j. $$ Let's see, multiplication is associative. define $$ \bar q = w - xi - yj - zk. $$ Oh, the norm of $q$ is $$ q \bar q = w^2 + x^2 + y^2 + z^2 $$ Now, take, as a "vector" $$ u = 2 i + j + k $$ of norm $6.$ There should be 24 possible vectors of norm $6.$
Finally, let $$ d = q \bar q \; . \; $$ The cute part is that $$ v = \bar q u q = ai + bj + c k $$ is another vector, no real part, and $$ a^2 + b^2 + c^2 = 6 d^2 $$ This way, we can deal with $d \equiv 7 \pmod 8,$ which cannot be written as the sum of three integer squares.
See what you can do with it. Note that, when $$ \gcd(a,b,c,d) = 1, $$ first $d$ is odd, then $a,b,c$ cannot be divisible by $3.$
$$\color{magenta}{ a = ( w+x+z)^2 + ( -w + x + y)^2 - 3 y^2 - 3 z^2 } $$
$$\color{magenta}{ b = ( w + x - 2 z)^2 - 2 ( x-y-z)^2 + 3 ( y+z)^2 - 6 z^2 } $$
$$\color{magenta}{ c = ( w - x + 2 y)^2 - 2 ( x-y-z)^2 - 3 ( y-z)^2 + 6 z^2 } $$
$$\color{magenta}{ d = w^2 + x^2 + y^2 + z^2 } $$
More later....1:45 pm my time.
Programmed it; very repetitive, as usual. The first thing I did was do a raw search, $a^2 + b^2 + c^2 = 6 d^2,$ with $\gcd(a,b,c) = 1,$ finally $a \geq b \geq c \geq 0.$ In that list, there are just three ways to get $d=7,$ just six ways to get $d=15.$