Sorry, I was unable to follow much of your post due to the fact that I'm still living in an ASCII world. --- Just think of me as a 'steampunk' mathematician channeling Hamilton/Tait/Maxwell/Sylvester, except that I have access to a computer running a symbolic algebra system (Maxima) which enables me to handle large equation sets impossible for a 19th C. denizen. I'm trying to imagine what these folks would have accomplished if Babbage had been successful and had produced Maxima. I.e., what if I had slept through the entire 20th C. of Bourbaki mathematics ? At 06:39 PM 10/13/2020, M F Hasler wrote:

Le mar. 13 oct. 2020 à 17:17, Dan Asimov <dasimov@earthlink.net> a écrit :

...

I skipped over sections of this post by Henry Baker to try to find the points that it included. But a general quaternion is *not* a product of quaternions that have zero j components and zero k components.

There seem to be many more or less original compilations of formulas about calculations in quaternions around on the web and on this list.

But there is the century old formalism of "standard" decompositions of complex 2x2 matrices into trace and traceless (aka "vectorial") part, based on standard representation theory.

(Clebsch-Gordan, Young tableaux, ...)

Those who have worked with Lie algebras and/or relativity willl probably know well the Clifford algebra of sigma or Pauli matrices:

any 2x2 matrix can be written as A = a⁰ s⁰ + *a* • *σ* *a* •â€¢ *σ* = a¹ σ¹ + a² σ² + a³ σ³ where σ⁰ = *I* is the identity matrix and *σ* =* т *= ( т¹ , i є , т³ ) are the three hermitian traceless Pauli matrices verifying σ¹ σ² = i ε¹²³ σ³ where indices 123 can be replaced by any indices *i,j,k* and ε is the completely antisymmetric tensor with ε¹²³ = 1.

(I use boldface for vectors since I can't put an arrow over them, I hope this comes through the mailing list... otherwise, as a rule of thumb, any lowercase letter without an index will represent a 3-vector (possibly of matrices, for σ & т) except for i and є .)

Then: A is hermitian iff (a⁰, a¹, a², a³) are all real ; det A = (a⁰)² - *a*² is the Minkowski "norm" of the 4-vector (a⁰,*a*) ; and a product is given by A B = (a⁰ b⁰ +* a *• *b*) σƒâ° + (a⁰* b* + b⁰* a* + i* a* × *b*) • *σ* wheree the last part involving the cross product, i (*a* × *b*) • *σ *is also the normalised commutator [A,B] = ½½(AB ­ BA).

Many more very nice formulas are easily disccovered, e.g. to extract components using the normalized trace: a¹ = ½ tr( A σ¹ ) and the same for all other indices 0 .. 3.

These formulas allow very easy calculations for quaternions, too, possibly using rather antihermitian matrices ­ *i* *σ**, * to "absorb" the imaginary unit in the structure constants (product & commutation relations), for the more popular (standard(?)) quaternion basis *i, j, k.*

(Where this boldface *i* obviously is not the imaginary unit!) - Maximilian

PS: Apologies to GR professionals who will prefer 4-vector index notation, use also 2-index (symmetric/traceless) sigma matrices, raise & lower 4-indices with eta = (-1,1,1,1) (or opposite according to religion) and matrix indices with the 2x2 epsilon tensor and its inverse, converting traceless <-> symmetric and scalar <-> antisymmetric.