RCHO Unification: Exploring Normed Division Algebras

[arivero]

http://arxiv.org/abs/1002.1497 :tongue: :rofl: :rofl: I just mentioned RCHO in other thread, and an article appears. Very predictive, I am :-DDDI haven't read it. I will not, most probably, in a few days. But I feel it could be place for a thread on the topic of relationship between unification and normed, division, algebras.

You may know, or perhaps not, that it is actually a *mainstream* topic. Evans did a proof of the relationship between supersymmetry and this kind of algebras. Of course it is also a "lost cause". But perhaps it could be regained. Also the vector-"diagonal" generalisation of Evans argument builds the full BraneScan, which can be also told to be mainstream. I told of the brane scan here, https://www.physicsforums.com/showthread.php?t=181194

BTW, the guy is at Perimeter, with Sorkin.

 

[arivero]

OK I read it. Well, the references DD Hey, I bet it is the first Perimeter preprint actually quoting F.D.T. Smith in the references, is it? No reference for my boss, althought :(

I am going to put some references and then I get the excuse to quote Boya too ;-)

Baez discussion, last year.
http://golem.ph.utexas.edu/category/2009/03/index_juggling_in_superyangmil.html
and Huerta homepage:
http://math.ucr.edu/~huerta/
Tony smith holistic webpage:
http://www.valdostamuseum.org/hamsmith/
Evans paper, you can download the preprint from HEP
http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B298,92
Spires search by title
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+T+DIVISIO%23+ALGEBRAS+OR+T+NORMe%23++ALGEBRAS&FORMAT=www&SEQUENCE=citecount%28d%29
Subtopic "octonions"
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+t+octonio%23+or+k+octonion%23&FORMAT=www&SEQUENCE=citecount%28d%29
RCHO as seen from Zaragoza
http://arxiv.org/pdf/hep-th/0301037v1

I should add some references to minor related topics, as for instance S7 and S13 and the Atiyah Arnold etc results on related fiberings there. But I refrain, hoping that some reader will also show interest on the topic...

 

[humanino]

You must be moving at quite some speed wrt to PF server, because a few days for you turned out to be 5 minutes here ! Are you running in circle ?

 

[arivero]

You must be moving at quite some speed wrt to PF server, because a few days for you turned out to be 5 minutes here ! Are you running in circle ?

Not circle, but U(1) :D Really I have not read it yet, only look the references. I happen to be busy on Hopf fiberings nowadays, so it was not a good idea to leave it go.
Ok, I am going to send it to the printer, to read in the bus.

 

[arivero]

It is not strange to get the standard model out from the octonions. The space of unit octonions is the sphere S7. And this sphere is "branched covered" by Witten's manifolds in a peculiar way:

S3 ---> S7 -----> S4 is the (generalised) Hopf fiber bundle of the sphere
S3 ---> M ------> CP2 is the fiber bundle schema both of Witten Manifold and also of some Aloff-Wallach spaces. In the first case, both the fiber and the bundle provide symmetries: SU(2)xU(1) and SU(3), respectively, as isometries of each. Remember S4 is HP1.

You can also fiber CP2 with an extra U(1) to get S5, whose isometry group is SO(6)=SU(4). This is the "lepton as the fourth colour" approach, and probably is nearer of Furey, who looks for the gauge group in C \otimes O.

The real problem of the RCHO approaches is to get the Higgs. Alain Connes got near of it, by considering two actions by CxH and CxM3, the 3x3 matrix, instead of O. See the Red Book.

If you don't get the Higgs, another mechanism for symbreak should exist. For instance, deformations of the metric. It is interesting that Alof-Wallach spaces do not have SU(2) isometry.

The relationship between CP2 and S4, as well as other RCHO compositions, is explained in math/0206135 by Atiyah and Berndt.

 

[MTd2]

I saw that paper, but I didn't read it. Why was that put on general physics? At first site, it looks like a serious work.

 

[arivero]

Let me go back to the unit sphere of octonions, I mean $S^7$. Note $S^3, S^1, S^0$ are groups, while $S^7$ is not. First clue that we are hitting the octonionic world.

A point that intrigues me is that we can see this sphere in four different spaces: $R^8, C^4, H^2, O$. Just as the $S^3$ can be seen in $R^4, C^2$ or $H$.

For $S^3$, the Hopf fibration works by projecting $C^2$ in $CP^1$.
For $S^7$, the Hopf fibration works by projecting $H^2$ in $HP^1$

But there are other projections. In $S^3$ I can also project in $RP^3$. In $S^7$ I can project in $RP^7$ or also in $CP^3$.

But I can not do "middle way" projections. I can not project $S^7$ in, say, $CP^2$ and get a meaningful fiber bundle. If I build U(1) fiber bundles over $CP^3$ I get the sphere again, but if I build U(1) fiber bundles over $CP^2 \times CP^1$ I get Witten's spaces, the ones with isometry group SU(3)xSU(2)xU(1).

It should be interesting to understand this game algebraically, down from the octonion sphere.

 

[arivero]

Some of this thread has been continued in this one
https://www.physicsforums.com/showthread.php?t=438585

 

[MTd2]

I told Tony Smith about the Octonion`s paper. He got really excited.

 

[MTd2]

Well, quoting from the other thread :)

It sound good because the space of unit octonions is S7, so back to 11 space time dimensions :-)

But the belt trick is in 3 spatial dimensions and it doesn`t seem that Furey defines anything in other than 3+1d. How come?

 

[arivero]

Well, quoting from the other thread :)
But the belt trick is in 3 spatial dimensions and it doesn`t seem that Furey defines anything in other than 3+1d. How come?

Just a minor correction here: it is Kauffman, no Furey, who does the belt trick in the last paper.

The important point, at the end, is if they can bypass Salam's objection about the charges. I guess that the C in CxO has a role there, because Bailin and Love did the bypass by going up one dimension.
And of course, they should solve the issue of breaking a SU(2)xSU(2) into SU(2)xU(1), perhaps related to chirality, and the selection for colour of SU(3) instead of SO(5). I think that these details are minor, but people will consider them important.

 

[MTd2]

Just a minor correction here: it is Kauffman, no Furey, who does the belt trick in the last paper.

I was talking about both of them. Kauffman in 3d and Furey in 3+1.

page 2 here:

"The groups are uni ed with the vectors they transform, and further, those vectors: the scalars, spinors, 4-momenta and eld strength tensors, are all born from the same meagre algebra."

http://www.perimeterinstitute.ca/personal/cfurey/UTI20100805.pdf

 

[arivero]

I was talking about both of them. Kauffman in 3d and Furey in 3+1.

page 2 here:

"The groups are uni ed with the vectors they transform, and further, those vectors: the scalars, spinors, 4-momenta and eld strength tensors, are all born from the same meagre algebra."

http://www.perimeterinstitute.ca/personal/cfurey/UTI20100805.pdf

Thers is no belt trick there. It is not that it can not be performed, but Furey tells nothing about it, it just puts a generic reference to Hestenes due to the use of Lorentz group. You have been dreaming some extra pages in the article, it seems :-)

Moreover, that parragraph refers to C \otimes H. Even if there were a belt trick in the references, it would not be about octonions.

 

[MTd2]

I wasn't talking about belt trick on Furey's paper. 3d for belt trick and Kauffman's paper and 3d+1 for Furey's.

You asked me about having a line of research for a new kind of quantum gravity. The paper mentions that, but I forgot where. I found it now:)

page. 10

"Extending on the relationship of the quaternions with SU(2) is the question of whether this model could provide illumination to attempts to use the octonions to construct the standard model of particle physics - such as the attempt in [2]. Here again the resemblance of L to parity inversion is suggestive of something more profound. We will continue these considerations in a sequel to the present paper"

Anyway, if you are a beginner, and like you said before, a very courageous one, and you receive and invitation or sugestion from a Grand Master like Louis Kauffman, wouldn't you follow it?

 

[arivero]

Ah, but Furey's is not 3+1.

He takes R,C,H,O, then he mixes a bit the R and the C, then he uses CxH to generate the Lorentz Group (acting in 3+1) and _separately_ he uses CxO to generate the gauge group... and he does not tell where this gauge group is acting. But if you compare CxH and CxO, you should deduce that the group got from CxO is acting also in some space. You could expect to be an object of a dimension 7+1. Which is the right result, because 7 is the minimum for the standard model and 8 for the GUT groups, and probably it is something between, because he should use R and C instead of two times C, when putting all together.

Note that in the standard theory also the 11 dimensional space divides in a very natural way into 4+7. This is well known.

 

[MTd2]

Let's see how many dimensions this RCHO has.

On page 2:

"The generic element of CHO is [FORMULA]. Imaginary units of the diferent division
algebras always commute with each other; explicitly, the
complex i commutes with the quaternionic i; j; k, all four
of which commute with the octonionic feng."

That means 2x4x8=64 dimensions.

Everything there happens in a certain hypersurface with the property of being and ideal of the algebra.

 

[arivero]

Let's see how many dimensions this RCHO has.

On page 2:

"The generic element of CHO is [FORMULA]. Imaginary units of the diferent division
algebras always commute with each other; explicitly, the
complex i commutes with the quaternionic i; j; k, all four
of which commute with the octonionic feng."

That means 2x4x8=64 dimensions.

Everything there happens in a certain hypersurface with the property of being and ideal of the algebra.

I count the unit ball in C times the unit ball in H times the unit ball in O. That makes 11.

Consider the product of a line and a circle. It is a cylinder. When you multiply manifolds, the dimensions add. And the same happens when you multiply commutative algebras (eg, the algebra of complex functions over the circle times the algebra of complex functions over the line): the dimensions of their Gelfand Naimark dual add.

A different problem is when you have a non commutative algebra. In this case two approaches are known: Morita equivalence, which doest a sort of reduction of most finite algebras to be equivalent to the algebra of complex functions over a dot. Or group theory, where you look for symmetry groups, for instance isolating the unit ball: a circle, a S3 sphere, or a S7 sphere for the respective case C, H, O.

 

[arivero]

This discusion is suggesting me an off-topic question. Suppose that you have an abstract theory that can be (Right/Wrong) and from the point of view of mainstream research it is (Orthodox/Heterodox).

Which is the order of preference you would assign to the four possible combinations?

I would go by RO > RH > WH > WO, but I am not militant.

 

[arivero]

You asked me about having a line of research for a new kind of quantum gravity. The paper mentions that, but I forgot where. I found it now:)

page. 10

"Extending on the relationship of the quaternions with SU(2) is the question of whether this model could provide illumination to attempts to use the octonions to construct the standard model of particle physics - such as the attempt in [2]. Here again the resemblance of L to parity inversion is suggestive of something more profound. We will continue these considerations in a sequel to the present paper"

I agree that the point raised in this parragraph is of grave consequence, seriousness or importance.

 

[MTd2]

Consider the product of a line and a circle. It is a cylinder. When you multiply manifolds, the dimensions add. And the same happens when you multiply commutative algebras (eg, the algebra of complex functions over the circle times the algebra of complex functions over the line): the dimensions of their Gelfand Naimark dual add.
.

There is no topological consideration on this paper. All there is direct multiplication of matrices. On the conclusion:

"Conclusion. Uni ed Theory of Ideals puts forward the
idea that all of the particles of the Standard Model, and
their transformations, come from a single algebra acting
on itself. This more powerful form of uni cation aims to
describe all of the gauge and spacetime degrees of freedom,
using only the 32 complex-dimensional algebra of
RxCxHxO".

So, 64 dimensional matrices.

 

[arivero]

C'mon, when someone shouts O, everyone hears S7. Furey doesn't need to waste a page of a good paper repeating such.

There is no topological consideration on this paper.

And thus no consideration about the dimension of space-time :-D. The dimension of a manifold is topology. You started to argue about 3+1, and then I went into a walk about the different ways to get some dimension given an algebra.

The number of components of the basis of the algebra is related to the representations of some group of symmetry (eg the dirac matrices are related to the 1/2,0+0,1/2 representation of Lorentz group), and this group is the group of isometries of a space (eg the Lorentz group recovers the usual space time)

Have you read this thread from the start? Lately I find that people -it applies even to myself- is very busy and has no time to read all the mail.

 

[MTd2]

There are 32 independent complex coeficients of RCHO, is should be a complex 32d space. I don`t think he is multiplying algebras, but using algebras elements as the basis of a space, s e(total)=e(basis for complex)x(basis for quaternions)xe(basis for octonions). So, the number of coeficients is 2x4x8=64 real or 32 complex.

There is 3+1 dimensions, but I think it is a statement about energy momentum vector, not the topology itself.

 

[MTd2]

I guess I know what you mean.

H X O= (1,J,K,L,e1,e2,e3,e4,e5,e6,e7) - 11d space

For me:

H X O = ( 1 X O, J X O , K X O, L X O) = (e1,e2,...,e7,Je1,Je2,...,Le7) - 32d space.

 

[arivero]

I guess I know what you mean.

H X O= (1,J,K,L,e1,e2,e3,e4,e5,e6,e7) - 11d space

For me:

H X O = ( 1 X O, J X O , K X O, L X O) = (e1,e2,...,e7,Je1,Je2,...,Le7) - 32d space.

But to get the 3+1 dimensions you are using the first rule, not the second one. CxH, according yourself, is 8 elements, not 3+1.

 

[MTd2]

I guess I see where the problem lies. The point is that combinations of the algebras RCHO, with different multiplication rules are used to find Ideals. For example, in the paper, 3 different multiplication rules are used with CXH to find 3 different quantities. Furey describes that in a graph.

 

[MTd2]

In the most updated paper, Furey wrote on the conclusion:

"Apart from this current work, Seth Lloyd is leading
the development of the theory of Division Networks, a
model for quantum gravity in the form of a lattice gauge
theory, which is written in the Unied Theory of Ideals
formalism."

http://www.perimeterinstitute.ca/personal/cfurey/UTI20100805.pdf

So, we can see a new quantum gravity theory emerging.

 

[arivero]

"a model for quantum gravity in the form of a lattice gauge theory, which is written in the Unied Theory of Ideals formalism."

So, we can see a new quantum gravity theory emerging.

It is more of a new, refreshing view, of an old approach, and really it was the most promising one. There are two problems to be solved:

- The double role of octonions (or division algebras in general): on one side they fix, via the brane scan the dimension for supersymetry, which is 10 for string theory and 11 for membrane theory. On other hand, they have a role in the symmetries of the 7 dimensional manifold. It is not known, afaik, how these roles are related.

- The role of chirality, with both its space-time aspect and its group theory aspect (in the electroweak group). The question is why the gauge theory group is neither SO(8) nor SO(5)xSU(2)xSU(2), which were the most likely candidates from the point of view of seven-dimensional geometry. Instead, the S4 sphere (with symmetry SO(5)) suffers some double-covering complex game and becomes CP2 (with symmetry SU(3)), at the same time that the S3 sphere (of SU(2)xSU(2) symmetry) falls into their lens space generalisation, any L3(q), and then the symmetry reduces to SU(2)xU(1). But in the usual interpretation, everything is still non-chiral. In this game Furey enters by telling that "Here again the resemblance of L to parity inversion is suggestive of something more profound. "

 

[MTd2]

- The double role of octonions (or division algebras in general): on one side they fix, via the brane scan the dimension for supersymetry, which is 10 for string theory and 11 for membrane theory. On other hand, they have a role in the symmetries of the 7 dimensional manifold. It is not known, afaik, how these roles are related.

I cannot understand what Furey's paper would have anything to do with string theory. There is no superpartner in the paper.

 

[arivero]

I cannot understand what Furey's paper would have anything to do with string theory. There is no superpartner in the paper.

Great point! Me neither

Seriously, it is a great mystery. We have by now a handful of high quality papers that happen to land in 6 or 7 extra dimensions without any mention of supersymmetry nor string theory. Between them, the landmark ones are Witten's Kaluza Klein and Connes's 2006 NCG model.

Near of these papers, we have some other work showing that the construction of 7 dimensional objects and its deformations are related to Hoft fiberings, and then to R, C, H, O. A lot was discovered by extending work done on Witten's KK, another pieces are pure math.

Then we have susy (but no superstring) gravity theories, sugra, that prefer to live in 6 or 7 extra dimensions.

Then we have Evan's work, proving that supersymmetry only works when there is a division algebra related to it, marking 3,4,6,10 as the only spacetime dimensions where susy can exitst.

Then we have, in original superstring theory, the critical spacetime dimension 10.

And finally we have the Brane scan, showing that the division algebras generate not only point supersymmetry in the R,C,H,O dimensions, but also a susy on extended objects, up to D=11.


A lot of theoretical lines pointing to the same extra dimensions! In fact, string theoretists are the one who worked *against* this research, because they prefer to extract the gauge group from a gauge group already in the string, instead of using the Kaluza Klein approach. The reason they argue: the failure of chirality in KK theories.

 

[MTd2]

It seems that with Furey's model, we just have to find ideals from combinations or R,C,H,O for a given multiplication rule. This is weird. Sounds like finding the spirit of finding an equation of motion for Lagrangian in the sense that both finds the extremum of an input function.

So, do we have a third energy equation. 1. Lagragian 2.Hamiltonian 3. Ideal ?

 

[kneemo]

Greetings Alejandro and MTd2 :)

Indeed, this recent work out of Perimeter involving the octonions is quite interesting.

Acquiring a full generation from $$\mathbb{C}\otimes\mathbb{O}$$ (i.e. the bioctonion algebra) goes back to the work of Gursey (reference [7] in Furey's paper), with further work being done by Catto, who showed the bioctonions give rise to a non-associative grassmann algebra and used their 3x3 Jordan algebra in an E6 unification model (http://arxiv.org/abs/hep-th/0302079" ). As a finite-dimensional composition algebra over $$\mathbb{C}$$, the bioctonions are maximal, as Springer and Veldkamp have shown the following:

Theorem
A finite dimensional vector space $$V$$ over a field $$\mathbb{F}=\mathbb{R},\mathbb{C}$$ can be endowed with a composition algebra structure if and only if $$\mathrm{dim}_{\mathbb{F}}(V)=1,2,4,8$$.Note: A composition algebra is an algebra $$\mathbb{A}=(V,\bullet)$$ admitting an identiy element, with a non-degenerate quadratic form $$\eta$$ (norm) satisfying

$$\forall x,y\in\mathbb{A}\quad\eta(x\bullet y)=\eta(x)\eta(y)$$.​

Over the reals, it turns out there are two non-isomorphic dimension eight composition algebras: the octonions and the split-octonions. Over the complex numbers, for a given dimension all composition algebras are isomorphic. The split-octonions underlie supergravity theories arising from toroidal compactifications of M-theory (most famously in N=8 supergravity in D=4), while the ordinary octonions so far have no direct interpretation in M-theory.

It is, however, quite easy to show the octonions and split-octonions are real subalgebras of the bioctonion algebra. In this sense, they are unified, and this unification can be uplifted to their corresponding 2x2 and 3x3 Jordan algebras, which are subalgebras of the 2x2 and 3x3 Jordan algebras over the bioctonions. The 3x3 Jordan algebra over the bioctonions was studied by Kaplansky and Wright http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001946" and is called the exceptional Jordan C*-algebra. This algebra played a pivotal role in proving that each Jordan-Banach algebra (JB-algebra) is the self-adjoint part of a unique Jordan C*-algebra.

Earlier this year, I used the exceptional Jordan C*-algebra and its Freudenthal triple system (FTS) to study extremal black holes in homogeneous supergravities based on the octonions and split-octonions http://arxiv.org/abs/1005.3514" . It was shown the bioctonions are essential in the study of M-theory on an 8-torus, which gives a D=3 supergravity with E8(8) U-duality group. Their utility arises in constructing the 57-dimensional space for which E8(8) is non-linearly realized, as the norm form on this space contains complex light-like solutions. This ultimately forces one to use the FTS over the bioctonions to complexify the 57-dimensional space, giving a realization of complex E8 on this space in which all light-like solutions are contained.

So what does this mean physically? Well, for one it hints at a new theory in which the split-octonion supergravity theories (e.g. N=8, D=4 SUGRA) arising from M-theory compactified on k-dimensional Tori are unified with the octonionic "magic" supergravity theories studied by Ferrara and Gunaydin http://arxiv.org/abs/hep-th/0606211" (which as of yet have no M-theory interpretation). If we are lucky, it might also shed some light on the proposed finiteness of N=8 supergravity.

 

[MTd2]

I will reproduce here part of an email exchange I had with Cohl.

******

Dear Cohl Furey,

I was thinking about your paper, with some colleagues of mine, and we came up with a sugestion to try find out the 3 generations.

Here's your paper:

http://www.perimeterinstitute.ca/personal/cfurey/UTI20100805.pdf

To find the fermions, you used the first formula for CXH, generalized for octonions:1. v=av' (page 2). So, you find the 1st generation fermions.
Why not using the other equations, (2) v=av'a+ and (3) v=av'ã? There are some compelling reasons for using them to find 3 generations. But with a few differences. For (3) we will use HX(CXO), instead of CXH

First, notice that (3) gives two ideals. A scalar and a tensor. The tensor part just uses the quaternion bases i,j,k. Similarly, one can do the same here and the result of the computation for fermions will be reused so that along i,j,k we have a generation. So, we have a tensor with 24x24 entries that gives the transition amplitude between the fundamental particles. This is a generalized CKM matrix or PMNS matrix http://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix

Using (2), we will find the a stronger version of the universality of the CKM matrix. (http://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix#Weak_universality)

What do you think? I would like your opinion.



Daniel.

*************

Hi Daniel,

Thanks for your email, and suggestion.

I will try to answer your questions:

> To find the fermions, you used the first formula for CXH, generalized for
> octonions:1. v=av' (page 2). So, you find the 1st generation fermions.
> Why not using the other equations, (2) v=av'a+ and (3) v=av'ã?

Good question. I have actually been working on getting gauge degrees
of freedom out of these other multiplication rules. I'm not sure it's
in the way that you mention, I'd be happy to let you know if I make
some progress on that front.

I'm not sure I understand your suggestion, could you clarify? Did you
mean to associate the quaternionic i with one generation, j with
another, and k with the 3rd? So that when you tensor that with CxO,
you get 3 copies of the single generation? (My apologies if I've
misunderstood.)

Best wishes,
Cohl
********************

Dear Cohl,

"I'm not sure I understand your suggestion, could you clarify? Did you
mean to associate the quaternionic i with one generation, j with
another, and k with the 3rd? So that when you tensor that with CxO,
you get 3 copies of the single generation? (My apologies if I've
misunderstood.)"

Yes, that`s it. And I forgot to mention, the scalar goes for the higgs, which is the 0th generation. There are other reasons to be that straightforward. The octonions live on the S7 sphere, whose group of symmetries is SO(8), so we have a triality relation in higher dimension between 3 preons whose extremities are tied to an S2 sphere. I say this assuming that you`ve read

http://arxiv.org/abs/1010.2979

And each preon is a buckle belt.

I guess it is not easy to know what is a gauge symmetry or a spatial symmetry, since all of this have complimentary description. I goes along your ideas.

Best wishes,

Daniel.

****************
Hi Daniel,

Interesting suggestion, I hadn't thought of that. So far I've been
trying to keep local spacetime degrees of freedom in CxH and internal
degrees of freedom in CxO, but as you mention, there is no compelling
reason right now to keep things separated in that way, apart from one
person's notion of aesthetics. Certainly Geoffrey Dixon didn't keep
things separated like that, and I would say if you think you see
something worth investigating, please, by all means write it up. I'm
very happy to listen.[...]
Cohl

 

[arivero]

Over the reals, it turns out there are two non-isomorphic dimension eight composition algebras: the octonions and the split-octonions. Over the complex numbers, for a given dimension all composition algebras are isomorphic. The split-octonions underlie supergravity theories arising from toroidal compactifications of M-theory (most famously in N=8 supergravity in D=4), while the ordinary octonions so far have no direct interpretation in M-theory.

Just to get a visualization: which is the topology of the unit ball of split-octonions? A 7-sphere?

 

[kneemo]

Just to get a visualization: which is the topology of the unit ball of split-octonions? A 7-sphere?

As the octonions and split-octonions have quadratic forms of signature (8,0) and (4,4) respectively, where the (4,4) signature gives rise to a pseudometric, the unit-"sphere" of the split-octonions resembles more a generalized 7D hyperboloid (or what some call a (3,4)-sphere http://www.hindawi.com/journals/amp/2009/483079.html" ).

 

[arivero]

As the octonions and split-octonions have quadratic forms of signature (8,0) and (4,4) respectively, where the (4,4) signature gives rise to a pseudometric, the unit-"sphere" of the split-octonions resembles more a generalized 7D hyperboloid (or what some call a (3,4)-sphere http://www.hindawi.com/journals/amp/2009/483079.html" ).

Interesting. Guess that it can be again connected to Hopf fibrations, but still it does not seem to be what we really need (S3xCP2).