Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$ Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
There is a parameterization for every equal sums of squares equation
$$
X_1^2 + \dotsb + X_m^2 = Y_1^2 + \dotsb + Y_n^2
$$
with $n,m$ positive integers and all $X_i,Y_i$ integers. The papers by Barnett and Bradley are my first recommendations.
Above equation (a^2+b^2+c^2)=(d^2+e^2+f^2) has parametric solution. 
Refer to Tito piezas on line book "collection of algebraic Identities".
Section sum of squares. The answer is given below;

(a,b,c)=[(p+q),(r+s),(t+u)]  and

(d,e,f)=[(p-q),(r-s),(t-u)]

Condition is (pq+rs+tu)=0

After parametrization of (pq+rs+tu)=0, the parametric solution in 
one variable is given below:

(a,b,c)=[(4k^2+12k+1),(3k-8),(7k+28)]    and

(d,e,f)=[(4k^2+12k-13),(k+2),(13k+26)]