The quaternion

-- -- | a b c d | | -b a -d c | | -c d a -b | | -d -c b a | -- --

The

You have probably already seen the corresponding result for complex numbrs:

If

If

Now,

Note that (exercise)

Suppose, first, that

(i) each of

(ii)

(ii) two of

In case (iii), we may relabel so that

Thus,

Note that not all of

Now

k^{2}mp | ||

= | (w^{2} + x^{2} + y^{2} + z^{2})
(s^{2} + t^{2} + u^{2} + v^{2}) | |

= | (ws+xt+yu+zv)^{2} +
(wt-xs+yv-zu)^{2} +
(wu-xv-ys+zt)^{2} +
(wv+xu-yt-zs)^{2} |

Again (exercise): each of

I have obviously leaned heavily on the excellent text by Hardy and Wright.

URL: http://math.fau.edu/locke/courses/ProblemSolving/Lagrange.htm