Euler's Four Square Identity

According to Euler's four square identity, the product of any two numbers a and b can be expressed as a sum of four squares if a and b both can individually be expressed as the sum of four squares.
Mathematically, if a = and b = 
Then, a * b = 
where c1, c2, c3, c4, d1, d2, d3, d4, e1, e2, e3, e4 are any integer.
 

Some examples are,

a =  = 30
b =  = 4
ab = a * b = 120 = 
a =  = 15
b =  = 24
ab = a * b = 810 = 
a =  = 15
b =  = 26
ab = a * b = 390 = 

Example: 

Input: a = 1 * 1 + 2 * 2 + 3 * 3 + 4 * 4
 b = 1 * 1 + 1 * 1 + 1 * 1 + 1 * 1
 
Output: i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0 * 0 + 2 * 2 + 4 * 4 + 10 * 10

i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2 * 2 + 4 * 4 + 6 * 6 + 8 * 8

Explanation : 
The product of the 2 numbers a(30) and b(4) can be represented as the sum of 4 squares as stated by Euler's four square identity. The above are the 2 representations of the product a * b in the sum of 4 squares form. All possible representations of the product a*b in the sum of four squares form are shown. 

Input: a = 1*1 + 2*2 + 3*3 + 1*1
 b = 1*1 + 2*2 + 1*1 + 1*1

Output: i = 0
j = 1
k = 2
l = 10
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 0*0 + 1*1 + 2*2 + 10*10

i = 0
j = 4
k = 5
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 0*0 + 4*4 + 5*5 + 8*8

i = 1
j = 2
k = 6
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 1*1 + 2*2 + 6*6 + 8*8

i = 2
j = 2
k = 4
l = 9
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 2*2 + 2*2 + 4*4 + 9*9

i = 2
j = 4
k = 6
l = 7
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 2*2 + 4*4 + 6*6 + 7*7

i = 3
j = 4
k = 4
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 3*3 + 4*4 + 4*4 + 8*8

Approach : 
Brute Force : 
A given number(a*b) can be represented in a sum of 4 squares form by using 4 loops i, j, k, l to find each of the four squares. This gives all possible combinations to form a*b as a sum of four squares. At each iteration of the innermost loop(l loop), check the sum with the product a*b. If there is a match, then print the 4 numbers(i, j, k, and l) whose sum of squares equals a*b. 
 

i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0*0 + 2*2 + 4*4 + 10*10

i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2*2 + 4*4 + 6*6 + 8*8

Improved Algorithm:
The time complexity of the above algorithm is in the worst case. This can be reduced to by subtracting the squares of i, j, and k from the product a*b for all (i, j, k) and checking if that value is a perfect square or not. If it is a perfect square, then we have found the solution. 
 

Output:  

 
i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0*0 + 2*2 + 4*4 + 10*10
i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2*2 + 4*4 + 6*6 + 8*8