Each pair of 4 integers sums to a square

How can I find a set of four integers such that the sum of any two is a square?

How can I find a set of four integers such that the sum of any two is a square.

Choose 6 integers, t1, t2, t3, s1, s2, s3.

Define
a1 = real part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) + i(t2-s2)) ((t3+s3+1) + i(t3-s3))

a2 = imaginary part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) + i(t2-s2)) ((t3+s3+1) + i(t3-s3))

a3 = real part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) + i(t2-s2)) ((t3+s3+1) - i(t3-s3))

a4 = imaginary part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) + i(t2-s2)) ((t3+s3+1) - i(t3-s3))

a5 = real part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) - i(t2-s2)) ((t3+s3+1) + i(t3-s3))

a6 = imaginary part of
((t1+s1+1) + i(t1-s1)) ((t2+s2+1) - i(t2-s2)) ((t3+s3+1) + i(t3-s3))

Define either,

p1 = (a1**2 + a3**2 - a6**2)/2
p2 = (a1**2 + a6**2 - a3**2)/2
p3 = (a3**2 + a6**2 - a1**2)/2
p4 = (a5**2 + a4**2 - a1**2)/2

or

p1 = (a1**2 + a3**2 - a5**2)/2
p2 = (a1**2 + a5**2 - a3**2)/2
p3 = (a3**2 + a5**2 - a1**2)/2
p4 = (a6**2 + a4**2 - a1**2)/2

according to which will make p1, p2, p3, p4 to be integers.
p1 + p2 = a1**2
p1 + p3 = a3**2
p1 + p4 = a5**2
p2 + p3 = a6**2
p2 + p4 = a4**2
p3 + p4 = a2**2
or
p1 + p2 = a1**2
p1 + p3 = a3**2
p1 + p4 = a6**2
p2 + p3 = a5**2
p2 + p4 = a4**2
p3 + p4 = a2**2
Numerical Example
Choose t1 = 2, t2 = 3, t3 = 5, s1=1, s2=1, s3=1
(4+i) (5+2i) (7+4i)
= ((4+i) (5+2i)) (7+4i)
= (18+13i) (7+4i)
= (74+163i)
a1 = 74
a2 = 163
(4+i) (5+2i) (7-4i)
= ((4+i) (5+2i)) (7-4i)
= (18+13i) (7-4i)
= (178+19i)
a3 = 178
a4 = 19
(4+i) (5-2i) (7+4i)
= ((4+i) (5-2i)) (7+4i)
= (22-3i) (7+4i)
= (166+67i)
a5 = 166
a6 = 67
a1**2 = 74**2 = 5476
a2**2 = 163**2 = 26569
a3**2 = 178**2 = 31684
a4**2 = 19**2 = 361
a5**2 = 166**2 = 27556
a6**2 = 67**2 = 4489
p1 = (a1**2 + a3**2 - a5**2)/2
p1 = (5476 + 31684 - 27556)/2
p1 = 4802
p2 = (a1**2 + a5**2 - a3**2)/2
p2 = (5476 + 27556 - 31684)/2
p2 = 674
p3 = (a3**2 + a5**2 - a1**2)/2
p3 = (31684 + 27556 - 5476)/2
p3 = 26882
p4 = (a6**2 + a4**2 - a1**2)/2
p4 = (4489 + 361 - 5476 )/2
p4 = -313
p1 + p2 = 5476 = 74**2
p1 + p3 = 31684 = 178**2
p1 + p4 = 4489 = 67**2
p2 + p3 = 27556 = 166**2
p2 + p4 = 361 = 19**2
p3 + p4 = 26569 = 163**2