By 'complex number' A, I sometimes mean the quaternion A+0j, constructed from the pair of complex numbers (A,0). This subgroup of the quaternions is isomorphic to the standard complex numbers. Yes, I'm abusing the notation, but mathematicians often abuse notation. :-) At 05:33 PM 10/13/2020, Dan Asimov wrote:

Henry, Despite whatever else you say, you also said: ----- We have thus factored our quaternion into two complex numbers sqrt(AB) and 1/sqrt(A'B), and a quaternion having only real (rather than complex) components. ----- As you have defined then, a "complex number" in the quaternions is a quaternion of form a + bi.

So maybe that's not what you mean, or maybe you don't mean that you have factored a quaternion into two complex numbers.

But those are things you said, and that was my point.

—Dan

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Hi Dan:

There's a lot of subtlety in my post that a quick scan might miss.

There's a *huge* difference between 'i' and 'j', so the eye might not pick up on this subtlety. In the quaternions, 'i' is perpendicular to 'j', and 'i' doesn't commute with 'j'. BTW, 'k' is there: it is simply represented by 'ij' rather than by a separate symbol.

My post is basically a rehash of the standard theorem about factoring SU(2) matrices into (diagonal complex)(real cos/sin)(diagonal complex) SU(2) matrices, but it does it completely within the quaternion framework.

At 02:16 PM 10/13/2020, Dan Asimov wrote:

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I skipped over sections of this post by Henry Baker to try to find the points that it included.

One point seems to be this: ----- We have thus factored our quaternion into two complex numbers sqrt(AB) and 1/sqrt(A'B), and a quaternion having only real (rather than complex) components. ----- But a general quaternion is *not* a product of quaternions that have zero j components and zero k components. So I not sure what this statement means.

—Dan