John Baez raises interesting points here that are somewhat peculiar to the 3rd & 4th dimensions given their relationship to the 4D quaternions. But there‘s a missed opportunity describing the symmetry in his way. It has to do with there being too many names for the 3D sphere S^3. https://twitter.com/johncarlosbaez/status/1290324611222011908">https://twitter.com/johncarlo...

You can call S^3 by all of the following names:

S^3, Spin(3), SU(2), Sp(1)

only they lead to *different* generalizations.

Unfortunately
Spin(4)=SU(2)xSU(2) is seldom written Spin(4)=Sp(1)xS^3 which should be preferred. But why is one naming better? It‘s because it generalizes.

The group Spin(8) is topologically also a product like Spin(4) of two spaces with algebraic product structures. Topologically, if we define Special Octonian groups so that Soct(1)=Spin(7), we have the following:

Spin(4)=Sp(1) x S^3
Spin(8)=Soct(1) x S^7

Raising a question.

While S^7 isn’t a group, it‘s an H-space w/ a multiplication rule.

Q1: How does Spin(8)’s associative Lie Group multiplication Arise from a non trivial non-product rule from its two topological H-Space factors when only one of them is associative.

Q2: Has this been done?

https://abs.twimg.com/emoji/v2/... draggable="false" alt="🙏" title="Folded hands" aria-label="Emoji: Folded hands">

Also, I understand that some people elsewhere on Twitter have been struggling against the brutal tyranny of:

2 + 2 = 4

Good luck to you. You will need it.