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The Quaternion identity expresses the product of any two numbers, each expressed as the sum of four squares, as the sum of four squares.

## Derivation from quaternionsEdit

The quaternion units 1,i,j,k are defined so that

1 = 1*1 = -i2 = -j2 = -k2
i = 1i = i1 = jk = -kj
j = 1j = -ik = j1 = ki
k = 1k = ij = -ji = k1

If u=a-bi-cj-dk, U=A+Bi+Cj+Dk, then from

|u||U|=|uU|,

we get

In defining u, the signs of b, c, and d are arbitrary; We could have just as easily defined them differently (or permuted the meanings of A, B, C and D). The important observations to make regarding the sign of the variables a,b,c,d,A,B,C,D on the right hand side of the equation are:

• One of the eight variables, which we will arbitrarily call a has a positive sign whereever it appears. This fixes a as one of the lower-case variables.
• One of the four variables upper-case variables, which we arbitrarily call A has a negative sign wherever it appears, except with a. This fixes the relationship between a and A.
• Then the pairing of b, c, and d with B, C, and D is such that bC, cD, and dB have a negative sign; and cB, dC, and bD have a positive sign.

## GeneralizationEdit

By defining $u=a-bi-\sqrt{n}cj-\sqrt{n}dk\,$ and $U=A+Bi+\sqrt{n}Cj+\sqrt{n}Dk\,$, then from

|u||U|=|uU|,

we get

(a2+b2+nc2+nd2)(A2+B2+nC2+nD2) =