Why not SEDENIONS?

Sedenions can be considered to be a ZeroDivisor Algebra.

Consider the following theorems (stated here without proof):

Generalized Frobenius: The alternative real division algebras, alternativity being defined in terms of the associator (x,y,z) = x(yz) - (xy)z by (x,y,z) = (-1)^P (Px,Py,Pz) for permutation P of x,y,z are: real numbers R; complex numbers C; quaternions Q; and octonions O.   In q-alg/9710013 Guillermo Moreno describes The Zero Divisors of the Cayley-Dickson Algebras over the Real Numbers. He shows, among other things, that the set of zero divisors (with entries of norm one) in the sedenions is homeomorphic to the Lie group G2. If the sedenions are regarded as the Cayley-Dickson product of two octonion spaces, then: if you take one 7-sphere S7 in each octonion space, and if you take G2 as the space of zero divisors, then YOU CAN CONSTRUCT FROM THE SEDENIONS the Lie group Spin(0,8) as the twisted fibration product S7 x S7 x G2. Such a structure is represented by the design of the Temple of Luxor.

Robert de Marrais has constructed detailed models for Zero Divisors of algebras of dimension 2^N, with explicit results through N = 8. He ( see his paper math.GM/0011260 ) visualizes the zero divisors of sedenions and higher-dimensional algebras in terms of Singularity / Catastrophe Theory, saying "... 168 elements provide the exact count of "primitive" unit zero-divisors in the Sedenions. ( [there are] Composite zero-divisors, comprising all points of certain hyperplanes of up to 4 dimensions ... ) The 168 are arranged in point-set quartets along the "42 Assessors" [in the Egyptian Book of the Dead the soul of the deceased passes through a hall lined by 21 pairs of Assessors] (pairs of diagonals in planes spanned by pairs of pure imaginaries, each of which zero-divides only one such diagonal of any partner Assessor). Wave-interference dynamics, depictable in the simplest case by Lissajous figures, ... derive from mutually zero-dividing trios of Assessors, a D4-suggestive 28 in number ... ". He also constructs "box-kites" that are similar to Onarhedra/heptavertons. Although he does not prove it in his paper, he conjectures that his box-kites are related to the H3 Coxeter Group Singularity and, that it, in turn, is related to the 16-dimensional Barnes-Wall lattice. I speculate that the H4 Coxeter Group Singularity, being based on the symmetries of the 600-cell, which in turn can be constructed from a 24-cell, may be related to the 24-dimensional Leech lattice.

Hurwitz:
The normed composition algebras with unit are:
real numbers;
complex numbers;
quaternions; and
octonions.

Dixon:
The algebras generated by sequences that are both
Galois sequences and
quadratic residue codes over GF(2) are:
real numbers   -   GF(1) and code length 0;
complex numbers -  GF(2) and code length 1;
quaternions   -    GF(4) and code length 3; and
octonions    -     GF(8) and code length 7.

Bott-Kervaire-Milnor:
(see Numbers, by Ebbinghaus, Hermes,
Hirzebruch, Koecher, Mainzer, Neukirch, Prestel,
and Remmert, Springer-Verlag 1991, including Chapter 11,
Division Algebras and Topology, by F. Hirzebruch;
and
Battelle Rencontres, ed. by Cecile M. DeWitt and
John A. Wheeler, W. A. Benjamin 1968, including Chapter XIX,
Continuous Solutions of Linear Equations - Some Exceptional
Dimensions in Topology, by Beno Eckmann)

By a theorem of Frobenius (1877), there are three
and only three associative finite division algebras:
1-dimensional real numbers R;
2-dimensional complex numbers C; and
4-dimensional quaternions Q.

By a theorem of Zorn (1933), every alternative,
quadratic, real non-associative algebra without
divisors of zero is isomorphic to the
8-dimensional octonions O.

In his paper, Fraleighs misstag: en varning,
Ernst Dieterich gives an example of a 4-dimensional
real division algebra that is NOT R, C, Q, or O,
even though it does have an identity 1 and a multiplicative inverse.
It is the vector space R x R3 with the product
(a,v)(b,w) = ( ab - v*w + v* Bx w , aw + bv + (By + Dd)(v x w) )
where * is transpose; a,b in R; v,w in R3; and (x,y,d) in K;
K = R3 x R3 x T; T = {d in R3| 0<d1<d2<d3}; and

0   -x3   x2                d1  0   0
B =  x3   0   -x1         Dd =   0   d2  0     (real entries).
-x2   x1   0                 0   0   d3

The value K = (0,0,(1,1,1)) gives the quaternions Q,
but for other values of the 9 parameters, it is not Q.
All members of the 9-parameter family are power-associative,
that is every subalgebra generated by 1 element is associative,
but only the quaternion algebra Q is alternative,
that is every subalgebra generated by 3 elements is associative
(in the case of Q, subalgebras generated by 3 independent elements
generate Q itself, so Q itself is also associative).
In 2 dimensions, every power-asociative real division algebra
is isomorphic to the complex numbers C.
In 8 dimensions, a construction similar to the one given above
for dimension 4, but for R x R7 using 7-dimensional vectors and
7x7 antisymmetic matrices gives a 21+21+7 = 49-parameter family
of 8-dimensional power-associative algebras that includes
as one of its members the alternative octonions O.

(You can use a Swedish Schoolnet web page to translate from Swedish to English.)
(My thanks to Daniel Asimov for telling me about Ernst Dieterich's work.)

In 1958, Kervaire and Milnor proved independently of each other
that the finite-dimensional real division algebras
have dimensions 1, 2, 4, or 8.
To do so, they used the periodicity theorem of Bott
on the homotopy groups of unitary and orthogonal groups.

In 1960, Adams proved that a continuous multiplication
in R(n+1) with two-sided unit and with norm product
exists only for n+1 = 1, 2, 4, or 8.

Ernst Dieterich says,
in an abstract of a talk on Real Quadratic Division Algebras:
"... [ R,C,Q,O ]... classifies all alternative real division algebras.
A natural further generalization to ask for is
the classification of all quadratic real division algebras.
Due to the famous (1,2,4,8)-theorem ... it suffices
to classify these in dimensions 4 and 8.
... Making systematic use of the basic general fact
that a quadratic algebra is `the same thing as'
an anti-commutative algebra endowed with a bilinear form (Osborn 1962),
we shall derive both
a complete classification of
all 4-dimensional real quadratic division algebras
and
the construction of a large family
of 8-dimensional real quadratic division algebras
which is conjectured to come close to a classification. ...", and
in a paper Eight-Dimensional Real Quadratic Division Algebras:
"... A real algebra A is called quadratic in case 0 < dim A,
there exists an identity element 1 in A and
each x in A satisfies an equation x^2 = ax + b1
with real coefficients a,b.
Every alternative real division algebra is quadratic.
The classification of all quadratic real division algebras
seems to be within reach. It is based on
their intimate relationship with dissident triples ...".
Examples of dissident triples are:
space R3, map Bx: R3/\R3 -> R,and dissident map (By+Dd): R3/\R3 -> R3
that gives
the 9-parameter family of 4-dimensional power-associative algebras;
and
space R7, map Bx: R7/\R7 -> R,and dissident map (By+Dd): R7/\R7 -> R7
that gives
the 49-parameter family of 8-dimensional power-associative algebras.

As Dave Rusin says on his web site:
"... A division ring need not have an identity by our definition
(although some authors require it). An example is
the set of complex numbers with
ordinary addition but with multiplication  z*w= z.conj(w)
(the complex conjugate of  w). ...
A nonassociative division ring  D  need not have an identity
element but is closely allied to one that does.
The appropriate connection is "isotopy":
every division algebra is isotopic to one with an identity. ...
So just how many distinct division algebras are there of dimension  n?
... the set of division algebras is an _open subet_ of  R^(n^3).
... the isomorphism classes of division algebras
... corresponds to the quotient of this space by the action of
GL(n,R) ... a Lie group of dimension  n^2, leaving a set
of dimension  n^3-n^2  to parameterize the set of all isomorphism
classes of division algebras.
For n=1, ...  R  is the only division algebra ...
For  n=2, ... there is a 4-parameter family of division algebras ...
For n=4: 48;
for n=8: 448. ...
On the other hand, many of these algebras are isotopic,
that is, the new product may be found from the old one as
x#y = P((Mx)(Ny)) where  M, N, and  P are fixed invertible matrices.
Roughly speaking,
this can provide at most a 3(n^2)-dimensional family of
framed division algebras starting from any single one.
... [all of the] 4-parameter family of division algebras of dimension 2
... are indeed ... isotopic to the complex field.
... [but] for n=4 or 8,
this procedure will not exhaust the set of division algebras
of dimension  n, so that there exist algebras which are non-isotopic,
and so more fundamentally different. ...".

In 1962, Adams proved that the maximum number of linearly
independent tangent vector fields on a sphere Sn is
equal to n for (n+1) = 2, 4, or 8,
and is less than n for all other n.
(It had been proven earlier that
the only parallelizable spheres are S1, S3, and S7.)

As Okubo has noted in his book
Introduction to Octonion and
Other Non-Associative Algebras in Physics
(Cambridge 1995), the theorem that
real division algebras must have dimension 1,2,4,8
"...has been derived on the basis of topological reasoning
on a seven-dimensional sphere.  A pure algebraic proof
of the theorem is still unknown."

However, a closely related theorem is the N-square theorem:

The product of two sums of N squares is a sum of N squares,
over the real numbers, if and only if N = 1, 2, 4, or 8.

There is a nice proof of the N-square theorem in the book
Squares, by A. R. Rajwade (Cambridge 1993).
Generalizing the N-square theorem, Rajwade defines
a triple of integers (r,s,n) as ADMISSIBLE over a field K iff
the product of a sum  of  r  squares and a sum of  s  squares
is a sum of  n  squares, over the field K.
The N-square theorem is just that (N,N,N) is admissible
over the real numbers for and only for:
(1,1,1), corresponding to 1-dim real numbers;
(2,2,2), corresponding to 2-dim complex numbers,
which are not ordered;
(4,4,4), corresponding to 4-dim quaternions,
which are not commutative; and
(8,8,8), corresponding to 8-dim octonions,
which are not associative.
(16,16,16) is NOT admissible, but
(16,16,32) is admissible over the reals.
Over the integral domain of the integers,
32 is the smallest n
for which (16,16,n) is admissible,
but over the real field it is only known
(as of the time of Rajwade's book)
that the smallest such n is
in the range from 29 to 32.

In light of the theorems, I think that Octonions are uniquely fundamental and I have used them to build the D4-D5-E6-E7 model.   However, many times I have been asked:   IF 8-DIM OCTONIONS ARE SO GOOD, WOULD NOT 16-DIM THINGS BE EVEN BETTER?   The 16-dim things that generalize octonions are called sedenions. The best reference for them known to me is NONASSOCIATIVE ALGEBRAS IN PHYSICS, by Jaak Lohmus, Eugene Paal, and Leo Sorgsepp Hadronic Press 1994, from which is taken the next figure, which uses notation {e0,e1,e2,e3,...,e15}.   A concrete example of zero divisors in terms of that basis is given by Guillermo Moreno in q-alg/9710013:   (e1 + e10)(e15 - e4) = -e14 - e5 + e5 + e14 = 0.   (See the multiplication table of Lohmus et al, below).       The basis used by Onar Aam is _ _ _ _ _ _ _ _ 1 i j k E I J K e i j k E I J K Sometimes I use a third notation, to avoid typing overbars:   1 i j k E I J K S T U V W X Y Z The real numbers R correspond to the empty set,
because R has no imaginary.
In all cases, the empty set corresponds to  1.

The complex numbers C correspond to a single point,
a 0-dimensional simplex,
because C has only one imaginary  i .

The quaternions correspond to a line segment,
a 1-dimensional simplex,
one end corresponding to             i ,
the segment corresponding to         j ,
and the other end corresponding to   k .
Going from i through j leads to k,
and so forth cyclically,
so that quaternion multiplication is determined
by the associative triple cycle

ijk

which gives quaternion multiplication rules
ii = jj = kk = ijk = -1 .
Since there was no triple cycle for the complex numbers,
the ijk associative triple cycle is new.
It corresponds to the Lie algebra Spin(2) = U(1).

The octonions correspond to a triangle,
a 2-dimensional simplex,
3 vertices corresponding to   I,J,K ,
3 edges corresponding to      i,j,k ,
and 1 face corresponding to     E .
There are 3+3+1 = 7 things.
There are 7 (projective) lines each with 3 things.
An octonion multiplication table,
one of the 480 that exist,
is then determined by the 7 associative triple cycles

ijk -iJK -IjK -IJk  EIi  EJj  EKk

where -iJK means that -iJK = -1.
For the octonions,
6 new associative triple cycles have appeared.
They correspond to the Lie algebra Spin(4).
The other 35 - 7 = 28 triples are not cycles.

The sedenions correspond to a tetrahedron,
a 3-dimensional simplex,
4 vertices v of the tetrahedron corresponding to  EIJK ;
6 edges e of the tetrahedron corresponding to    ijkTUV ;
4 faces f of the tetrahedron corresponding to     WXYZ ;
and the 1 entire tetrahedron T corresponding to    S  .
There are 4+6+4+1 = 15 things.
There are 35 (projective) lines each with 3 things.
Geometrically, they are of the form:
4  like eee (where eee are all on the same face);
6  like vev (these are the edges);
12 like vfe (where v is opposite e on face f);
3  like eTe (where e is opposite e on the whole tetrahedron T);
4  like vTf (where v is opposite f on T); and
6  like fef (where the edge of e is not on f or f,
that is, f and f are opposite to e).
A sedenion multiplication table
(there are even more than 480 of them)
is then determined by the 35 associative triple cycles
used by Onar Aam:

ijk -iJK -IjK -IJk  EIi  EJj  EKk

__   __   __   __   __   __   __
-ijk  iJK  IjK  IJk -EIi -EJj -EKk
_ _  _ _  _ _  _ _  _ _  _ _  _ _
-ijk  iJK  IjK  IJk -EIi -EJj -EKk
__   __   __   __   __   __   __
-ijk  iJK  IjK  IJk -EIi -EJj -EKk

__   __   __   __   __   __   __
eEE  eII  eJJ  eKK  eii  ejj  ekk

For sedenions,
28 new associative triple cycles have appeared.
They correspond to the Lie algebra Spin(0,8).
The other 455 - 35 = 420 triples are not cycles.

Lohmus, Paal, and Sorgsepp give a table for the
same sedenion multiplication product: In the 16x16 table,
the upper left 1x1 gives a table for R,
the upper left 2x2 gives a table for C,
the upper left 4x4 gives a table for Q,
and the upper left 8x8 gives a table for O.

Here is the same table in 1ijkEIJKSTUVWXYZ notation:

1  i  j  k   E  I  J  K    S  T  U  V   W  X  Y  Z

1    1  i  j  k   E  I  J  K    S  T  U  V   W  X  Y  Z
i    i -1  k -j   I -E -K  J    T -S -V  U  -X  W  Z -Y
j    j -k -1  i   J  K -E -I    U  V -S -T  -Y -Z  W  X
k    k  j -i -1   K -J  I -E    V -U  T -S  -Z  Y -X  W

E    E -I -J -K  -1  i  j  k    W  X  Y  Z  -S -T -U -V
I    I  E -K  J  -i -1 -k  j    X -W  Z -Y   T -S  V -U
J    J  K  E -I  -j  k -1 -i    Y -Z -W  X   U -V -S  T
K    K -J  I  E  -k -j  i -1    Z  Y -X -W   V  U -T -S

S    S -T -U -V  -W -X -Y -Z   -1  i  j  k   E  I  J  K
T    T  S -V  U  -X  W  Z -Y   -i -1 -k  j  -I  E  K -J
U    U  V  S -T  -Y -Z  W  X   -j  k -1 -i  -J -K  E  I
V    V -U  T  S  -Z  Y -X  W   -k -j  i -1  -K  J -I  E

W    W  X  Y  Z   S -T -U -V   -E  I  J  K  -1 -i -j -k
X    X -W  Z -Y   T  S  V -U   -I -E  K -J   i -1  k -j
Y    Y -Z -W  X   U -V  S  T   -J -K -E  I   j -k -1  i
Z    Z  Y -X -W   V  U -T  S   -K  J -I -E   k  j -i -1

Lohmus, Paal, and Sorgsepp note that if you use the Cayley-Dickson procedure to double the octonions to get the sedenions, you retain the properties common to all Cayley-Dickson algebras:   centrality if xy = yx for all y in the algebra A, then x is in the base field of A, which is the real numbers R; simplicity no ideal K other than {0} and the algebra A, or, equivalently, if for all x in K and for all y in A xy and yx are in K, then K = {0} or A; flexibility (x,y,z) = (xy)z - x(yz) = -(z,y,x) or, equivalently, (xy)x = x(yx) = xyx ; power-associativity (xx)x = x(xx) and ((xx)x)x = (xx)(xx) or, equivalently, x^m x^n = x^(m+n) ; Jordan-admissibility xoy = (1/2)(xy + yx) makes a Jordan algebra; degree two xx - t(x)x + n(x) = 0 for some real numbers t(x) and n(x) ; derivation algebra G2 for octonions and beyond; and squares of basic units = -1 .     For sedenions, you lose the following properties:   the division algebra (over R) property xy = 0 only if x =/= 0 and y =/= 0 ; (A concrete example of zero divisors in terms of that basis is given by Guillermo Moreno in q-alg/9710013: (e1 + e10)(e15 - e4) = -e14 - e5 + e5 + e14 = 0.)   linear alternativity (x,y,z) = (xy)z - x(yz) = (-1)P(Px,Py,Pz) where P is a permutation of sign (-1)P ;   and the Moufang identities (xy)(zx) = x(yz)x (xyx)z = x(y(xz)) z(xyx) = ((zx)y)x .     For sedenions, you retain the following properties:   anticommutativity of basic units xy = -yx;   and nonlinear alternativity of basic units (xx)y = x(xy) and (xy)y = x(yy).
The 28 new associative triple cycles of the sedenions are related to the 28-dimensional Lie algebra Spin(0,8), and to the 28 different differentiable structures on the 7-sphere S7 that are used to construct exotic structures on differentiable manifolds.   Topological study of such manifolds has produced the only presently known proofs that the dimension of a division algebra must be 1, 2, 4, or 8. No algebraic proof is now known (Okubo (1995)).   The only real Euclidean space with exotic differentiable structure is R4. In particular, R4 has many exotic differentiable structures # such that there exists a compact subset of any exotic R4# that cannot be smoothly embedded in ANY 3-sphere S3 in the R4#.   In gr-qc/9405010 (also see gr-qc/9604048 by Carl Brans and hep-th/9604137 by J. Sladowski ), Carl Brans has suggested that, instead of looking at R4#, remove a point to get a semi-exotic cylinder R1 x# S3 where x# denotes topological (not differentiable) product. Brans remarks that an exotic spacetime R1 x# S3 could not have smooth differentiable structure forever, but that at some point on the time R1 axis, an obstruction would be encountered.   Nobody knows what such an obstruction would look like.   Another unknown is whether or not there exists an exotic S1 x# S3. Since the 4-dimensional spacetime of the D4-D5-E6-E7 physics model is of the form RP1 x S3, with topological structure S1 x S3, and exotic S1 x# S3 might be of physical interest.   Before dimensional reduction, the D4-D5-E6-E7 physics model has 8-dimensional spacetime of the form RP1 x S7, with topological structure S1 x S7. Therefore, it is interesting to look at exotic structures on spheres, particularly S7.   The Milnor spheres S(4k-1)#, of dimension 4k-1 for k=2 or greater, are homeomorphic to normal spheres S(4k-1) but not diffeomorphic to them.   The 28 differentiable structures of the 7-sphere enabled John Milnor (Ann. Math. 64 (1956) 399) to construct exotic 7-spheres, denoted here by S7#. There are 27 exotic S7# spheres, plus one (the 28th) normal S7. A 7-sphere, whether exotic S7# or normal S7, can be "factored" by a Hopf fibration into a 3-sphere S3 and a 4-sphere S4. Each point of the S4 can be thought of as having one S3 attached. The Hopf fibration can be denoted by S3 - S7 - S4.   Hopf fibrations can only be done for spheres of dimension 1,3,7,15: S0 - S1 - S1 (S0 = point) based on real numbers S1 - S3 - S2 based on complex numbers S3 - S7 - S4 based on quaternions S7 - S15 - S8 based on octonions     Daniel Asimov has shown that the dimensions k = 0, 1, 3, and 7 are the only dimensions for which an open set in R^(2k+1) can be continuously filled by k-hoops.     Consider S3 - S7 - S4 based on quaternions   It is pretty clear that this is based on building the parallelizable S7, with coordinates ijkEIJK, from the associative ijk triple S3 and the coassociative EIJK square S4 of Onar Aam.   Since the 3-sphere S3 is parallelizable, we can use ijk as a tangent coordinate system at any point and carry it around to any other point, so that ijk is a global coordinate system for the S3.   Since the 4-sphere S4 is NOT parallelizable, if we put a tangent EIJK coordinate space at a particular point of the S4, we cannot carry it around to all other points of the S4.   What we must do to define a "tangent" EIJK space at every point of S4 is to split S4 into two hemispheres, north and south, say, and put one EIJK tangent space on the northern hemisphere and another EIJK tangent space on the southern hemisphere.   Notice that the two hemispheres meet at the equator. The equator of a 4-sphere is a 3-sphere.   Since both northern and southern hemispheres have EIJK coordinate tangent spaces, you can choose IJK as the tangent coordinates for each of them, so that they agree on their common points, the S3 equator.   This gives the normal 7-sphere S7, factored into ijk 3-sphere S3 and EIJK + EIJK 4-sphere S4 with IJK IJK equatorial S3 of S4.     How do we get the exotic 7-spheres S7#?   An S7# should be constructed by factoring into ijk 3-sphere S3 and STUV + WXYZ 4-sphere S4 with TUV XYZ equatorial S3 of S4.   where STUV and WXYZ are not the same, but they must be mathematically consistent in that if S is a mirror reflecting ijk and TUV then W must also be a mirror reflecting ijk and XYZ, with the same orientation.   How many different ways can this be done?   It amounts to how many ways you can choose a triple out of the 8 STUV WXYZ 8! / 3! 5! = 8x7x6 / 3x2 = 56 and then choosing an orientation (half of the 56) to get 56 / 2 = 28 ways.   IT IS ALSO EQUIVALENT TO ONAR AAM'S CONSTRUCTION OF 28 "NEW" triple cycles IN THE SEDENIONS.   Carl Brans in gr-qc/9404003 conjectures (in the context of exotic R4#) that localized exoticness can act as a source of a physical field. In gr-qc/9906037, J. Sladowski uses isometry groups including SO(n,1) and SO(n,2) to show the validity of a form of the Brans conjecture: there are four-manifolds (spacetimes) on which differential structures can act as a source of gravitational force just as ordinary matter does.   Since prior to dimensional reduction, the D4-D5-E6-E7 physics model has 8-dimensional spacetime of the form RP1 x S7 with topological structure S1 x S7, we can consider exotic structures on the spatial S7. As we just saw, there are 27 exotic S7# in addition to normal S7. If each Planck-size neighborhood of spacetime were a domain with its own S7-S7# structure, then each domain would act as a vertex in the 8-dim HyperDiamond lattice version of the D4-D5-E6-E7 model, and boundaries between domains would act as links. Since there are 28 different domain structures, isomorphic to 28-dim Spin(0,8), the boundaries/links would carry Spin(0,8) gauge bosons just as in the D4-D5-E6-E7 model prior to dimensional reduction.   What about after dimensional reduction? The S1 x S7 spacetime structure is effectively reduced to S1 x S3 spacetime structure (with unknown exotic structures) plus a 4-dimensional coassociative internal symmetry space. If the internal symmetry space is regarded as S4, it has no exotic structure, and all the exociticity of S1 x S7 must go to the S1 x S3. If the internal symmetry space is regarded as S1x S3 or as R1 x S3 or as R4, then the internal symmetry space may have some exotic structure.   This raises interesting questions whose answers may be useful in constructing physically realistic models.
SEDENIONS AND CLIFFORD ALGEBRAS:   If they do not look at the whole sedenion algebra, but represent sedenions by their left or right adjoint actions, When Lohmus, Paal, and Sorgsepp get interesting matrix structures.   To see how this works, first consider the octonion algebra:   Let x and X be octonions, and let * denote octonion conjugation.   Let Lx, *Lx, LX, and *LX be octonion left-actions.   Let Rx, *Rx, RX, and *RX be octonion right-actions.   As Dixon shows, the octonion left and right actions can be represented by 8x8 real matrices acting on the space of 1x8 real vectors, or the space of octonions.   Consider the 7 matrices representing the imaginary octonions. The anticommutator of any two of them {Lp,Lq} = - 2 DELTA(pq) so that the 7 matrices generate the 128-dimensional Clifford algebra Cl(0,7), whose even subalgebra is 64-dimensional, whose minimal ideal spinor space OSPINOR is 8-dimensional.   The 0-grade 1-dimensional scalar space of Cl(0,7) represents the octonion real axis.   There is a 1 to 1 correspondence between the 1x8 minimal ideal OSPINOR on which OL acts by Clifford action, and the 1x8 octonion column vectors on which OL acts by matrix-vector action.   This not only leads to triality in the larger Clifford algebra Cl(0,8) of Spin(0,8), but also to the division algebra property of octonions, because the map OL from OSPINOR to O is 1 to 1 and invertible.   The space OR of octonion right-actions is equal to OL.     Now - LOOK AT SEDENIONS:   Lohmus, Paal, and Sorgsepp define sedenion left-actions SL by a 2x2 matrix of 8x8 matrices, which is the 16x16 matrix:   OLx -*ORx *OLx ORx   where *OL is the conjugate of OL and *x is the conjugate of x.   They define sedenion right-actions SR by a 2x2 matrix of 8x8 matrices:   ORx -OL*x OLx OR*x   Thus, they represent the sedenion left and right actions SL and SR by 16x16 real matrices acting on 1x16 real vectors.   Consider the 15 matrices representing the imaginary sedenions. The anticommutator of any two of them {Lp,Lq} = - 2 DELTA(pq) so that the 15 matrices generate the 32,768-dimensional Clifford algebra Cl(0,15), whose even subalgebra is 16,384-dimensional, whose minimal ideal spinor space SSPINOR is 128-dimensional.   The 0-grade 1-dimensional scalar space of Cl(0,15) represents the sedenion real axis.   There is an 8 to 1 correspondence between the 1x128 minimal ideal SSPINOR on which SL acts by Clifford action, and the 1x16 sedenion column vectors on which SL acts by matrix-vector action.   This leads to failure of the division algebra property of sedenions, because the map SL from SSPINOR to S is 8 to 1 and invertible.
Consider SLx of sedenion left-multiplication by x as being represented by the 16x16 real matrix   OLx -*ORx *OLx ORx   and consider the 16x16 real matrices forming the 256-dim matrix algebra R(16), which is the Clifford algebra Cl(0,8) of Spin(0,8):   0 2 2 2 2 2 2 2 7 5 5 5 5 5 5 5 4 4 2 2 2 2 2 2 5 7 5 5 5 5 5 5 4 4 4 2 2 2 2 2 5 5 7 5 5 5 5 5 4 4 4 4 2 2 2 2 5 5 5 7 5 5 5 5 4 4 4 4 4 2 2 2 5 5 5 5 7 5 5 5 4 4 4 4 4 4 2 2 5 5 5 5 5 7 5 5 4 4 4 4 4 4 4 2 5 5 5 5 5 5 7 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 7 1 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 3 1 3 3 3 3 3 3 6 4 4 4 4 4 4 4 3 3 1 3 3 3 3 3 6 6 4 4 4 4 4 4 3 3 3 1 3 3 3 3 6 6 6 4 4 4 4 4 3 3 3 3 1 3 3 3 6 6 6 6 4 4 4 4 3 3 3 3 3 1 3 3 6 6 6 6 6 4 4 4 3 3 3 3 3 3 1 3 6 6 6 6 6 6 4 4 3 3 3 3 3 3 3 1 6 6 6 6 6 6 6 8   The numbers refer to the grade in Cl(0,8) of the matrix entry: grade 0 1 2 3 4 5 6 7 8 dimension 1 8 28 56 70 56 28 8 1       Important Notation Notes:   The two blocks of the form   0 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 are more symbolic than literal. They mean that: the 28 entries labelled 2 correspond to the antisymmetric part of an 8x8 matrix; the 35 entries labelled 4 correspond to the traceless symmetric part of an 8x8 matrix; and the 1 entry labelled 0 corresponds to the trace of an 8x8 matrix.   A more literal, but more complicated, representation of the graded structure of those two blocks is:   0 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4   However, in the more literal representation, the entries are not all independent. The more symbolic representation is a more accurate reflection of the number of independent entries of each grade.     The two blocks of the form   1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1   can be taken more literally, as they mean that: the 8 entries labelled 1 correspond to the diagonal part of an 8x8 matrix; and the 56 entries labelled 3 correspond to the off-diagonal part of an 8x8 matrix.   The conventions of the above Notation Notes are used from time to time in my papers and web pages.     The even subalgebra Cle(0,8) of Cl(0,8) is then the block diagonal     0 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 4 4 4 4 4 4 4 6 6 4 4 4 4 4 4 6 6 6 4 4 4 4 4 6 6 6 6 4 4 4 4 6 6 6 6 6 4 4 4 6 6 6 6 6 6 4 4 6 6 6 6 6 6 6 8   The SLx matrix action of sedenion left-multiplication by x restricted to the block diagonal of the even subalgebra Cle(0,8) is then   OLx ORx   and the block diagonal part of the SL matrices is just the direct sum OL + OR each of which is an 8x8 real matrix acts on 8-dimensional vector space isomorphically to its action of 8-dimensional spinor space OSPINOR.   Denote the OL spinor space by OSPINOR+ and the OR spinor space by OSPINOR-.   Then, the direct sum OSPINOR+ + OSPINOR- represent the +half-spinor space and the -half-spinor space of the Clifford algebra Cl(0,8) of Spin(0,8)   + + + + + + + +   - - - - - - - -   The +half-spinor space OSPINOR+ is acted on by the OL elements of Cle(0,8) of   grade 0 2 4 dimension 1 28 35   while   the -half-spinor space OSPINOR- is acted on by the OR elements of Cle(0,8) of   grade 4 6 8 dimension 35 28 1 we have the useful result that the block diagonal part of the adjoint left action SL of sedenions represents the 16-dimensional full spinor representation of the Clifford algebra Cl(0,8) of the Lie algebra Spin(0,8).