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

- (a
^{2}+b^{2}+c^{2}+d^{2})(A^{2}+B^{2}+C^{2}+D^{2}) = (aA+bB+cC+dD)^{2}+(aB-bA-cD+dC)^{2}+(aC+bD-cA-dB)^{2}+(aD-bC+cB-dA)^{2}

## Derivation from quaternionsEdit

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

- 1 = 1*1 = -i
^{2}= -j^{2}= -k^{2} - 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

- (a
^{2}+b^{2}+c^{2}+d^{2})(A^{2}+B^{2}+C^{2}+D^{2}) = (aA+bB+cC+dD)^{2}+(aB-bA-cD+dC)^{2}+(aC+bD-cA-dB)^{2}+(aD-bC+cB-dA)^{2}

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

- (a
^{2}+b^{2}+nc^{2}+nd^{2})(A^{2}+B^{2}+nC^{2}+nD^{2}) =

(aA+bB+ncC+ndD)^{2}+(aB-bA-ncD+ndC)^{2}+n(aC+bD-cA-dB)^{2}+n(aD-bC+cB-dA)^{2}

## ApplicationEdit

A simple application of this identity is

- 3(A^2+B^2+C^2+D^2) = (A+B+C)^2+(D+A-B)^2+(C+D-A)^2+(B-C+D)^2

## See alsoEdit

- Complex product identity, that the product of two numbers expressed as the sum of two squares can be expressed as the sum of two squares.