APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 69 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥⊃+/w=e×ばつ,(⍉↓r∘.+r∘.×ばつ32 8)×ばつe×ばつ⍨⍳w}

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^{-4 bytes thanks to @ngn and @H.PWiz.}

APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 67 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥+/w=e×ばつ⍪(r∘.+⍨32 ×ばつr)×ばつe×ばつ⍨⍳w}

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^{-6 bytes thanks to @ngn and @H.PWiz.}

APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 71 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥⊃+/w=e×ばつ,(×ばつ32 8)×ばつe×ばつ⍨⍳w}

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^{-2 bytes thanks to @ngn.}

f←{
 ⍝ Extract squarefree part of ⍵
 ×ばつ⍨⍳⍵
⍝ ×ばつ⍨⍳⍵ ⍝ Square of 1..⍵
⍝ ⍵|⍨ ⍝ Remainders of dividing ⍵ by each of the above
⍝ ⊃⌽⍒ ⍝ Last item of descending sort order; index of last 0
⍝ ×ばつ⍨ ⍝ Remove square term and name it w
 ⍝ 1 if odd, 2 if even; divide w by that
 w÷←e←2-2|w
⍝ e←2-2|w ⍝ 2 minus w modulo 2; name it e
⍝ w÷← ⍝ Divide w by e and overwrite to w
 ⍝ Square of plus-minus range
 ×ばつ⍨⍳w
⍝ ×ばつ⍨⍳w ⍝ Square of 1..w
⍝ r←0,2/ ⍝ Make two copies of each number, prepend 0 and name it r
 ⍝ y2+32z2 (A/C) and y2+8z2 (B/D), with y and z swapped
 m←∘.(32 8⊥ ̈∘⊂,)⍨r
⍝ ∘.( )⍨r ⍝ Self outer product of r by...
⍝ , ⍝ Concatenate two input numbers and
 32 8⊥ ̈∘⊂ ⍝ Interpret as base-32 and base-8
⍝ m← ⍝ Name it m
 ⍝ Add ×ばつe×ばつx2
 ×ばつe×ばつr
⍝ ×ばつe×ばつr ⍝ Outer product by + with ×ばつe×ばつx2
⍝ m←, ⍝ Flatten and overwrite m
 ⍝ count w in m and check if B/D is twice A/C
 0= ̄2⊥⊃+/w=m
⍝ w=m ⍝ Test for each sum being w
⍝ ⊃+/ ⍝ Count of 'w's in A/C and B/D
⍝ ̄2⊥ ⍝ ( ×ばつA/C) + B/D
⍝ 0= ⍝ Equals 0?
}

APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 69 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥⊃+/w=e×ばつ,(×ばつ32 8)×ばつe×ばつ⍨⍳w}

Try it online!

^{-4 bytes thanks to @ngn and @H.PWiz.}

f←{
 ⍝ Extract squarefree part of ⍵
 ×ばつ⍨⍳⍵
⍝ ×ばつ⍨⍳⍵ ⍝ Square of 1..⍵
⍝ ⍵|⍨ ⍝ Remainders of dividing ⍵ by each of the above
⍝ ⊃⌽⍒ ⍝ Last item of descending sort order; index of last 0
⍝ ×ばつ⍨ ⍝ Remove square term and name it w
 ⍝ 1 if odd, 2 if even; divide w by that
 w÷←e←2∨w
⍝ e←2∨w ⍝ GCD of w with 2; name it e
⍝ w÷← ⍝ Divide w by e and overwrite to w
 ⍝ Square of plus-minus range
 ×ばつ⍨⍳w
⍝ ×ばつ⍨⍳w ⍝ Square of 1..w
⍝ r←0,2/ ⍝ Make two copies of each number, prepend 0 and name it r
 ⍝ y2+32z2 (A/C) and y2+8z2 (B/D), with y and z swapped
 m←∘.(32 8⊥ ̈∘⊂,)⍨r
⍝ ∘.( )⍨r ⍝ Self outer product of r by...
⍝ , ⍝ Concatenate two input numbers and
 32 8⊥ ̈∘⊂ ⍝ Interpret as base-32 and base-8
⍝ m← ⍝ Name it m
 ⍝ Add ×ばつe×ばつx2
 ×ばつe×ばつr
⍝ ×ばつe×ばつr ⍝ Outer product by + with ×ばつe×ばつx2
⍝ m←, ⍝ Flatten and overwrite m
 ⍝ count w in m and check if B/D is twice A/C
 0= ̄2⊥⊃+/w=m
⍝ w=m ⍝ Test for each sum being w
⍝ ⊃+/ ⍝ Count of 'w's in A/C and B/D
⍝ ̄2⊥ ⍝ ( ×ばつA/C) + B/D
⍝ 0= ⍝ Equals 0?
}

APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 72 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥⊃+/w=e×ばつ,(∘.(32 8⊥ ̈∘⊂,)⍨r)×ばつe×ばつ⍨⍳w}

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^{-1 byte thanks to @ngn.}

APL (Dyalog Unicode), ^{(削除) 73 (削除ここまで)} 71 bytes

{e×ばつ⍨⍳⍵⋄0= ̄2⊥⊃+/w=e×ばつ,(⍉↓r∘.×ばつ32 8)×ばつe×ばつ⍨⍳w}

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^{-2 bytes thanks to @ngn.}