**The Octonions
Yet Another Essay
**

**
Table of Contents:**

Introduction

Getting Acquainted

An Action by the Dihedral Group D_{6}

The Fano Plane

The Exceptional Lie Algebra g_{2}

The Hamming Code and Gosset's Lattice

Integral Octonions

The Magic Square of Freudenthal

References

Here are some unapologetically rambling notes on octonions. I like to play with these things more than I probably should, so I have collected some scraps here and there and tried to assemble them into a cogent webpage. I don't know how you found this page, although I would guess it was through an internet search using the keyword "octonions". It's commendable that you are not wasting your time with video games, comic books, science fiction or romance novels, or blackjack, and even more interesting that you instead have the curiosity about the algebro-geometric jewels which are the octonions. Thank-you for allowing me to guide you through this fascinating realm. Use caution -- this stuff can be addictive and time-consuming, just like video games, comic books, etc.

The space of octonions, or "Cayley numbers" as many refer to them, or "octaves"
as their inventor John Graves called them, are apparently interesting for many
reasons.
Octonions seem to lie near the entrance to a museum of "exceptional" geometric objects
associated especially with the five exceptional complex simple Lie algebras.
It is probably impossible to give a complete survey of these kinds of results,
as new results seem to continue to arise. Using this page, if the
reader works a little, one can probably convince onself at least that (a)
the octonions provide an example of a non-associative algebra which is truly
not associative, (b) one can use the octonions to construct the 14-dimensional
simple Lie algebra g_{2}, (c) the octonions provide a convenient presentation
of Gosset's exceptional 8-dimensional lattice E_{8} as a (non-associative) ring, and
(d) if you are bothered, impatient, or otherwise upset about a lack of associativity, then at the very least
you might find the existence of octonions (and their relationship to geometry) somewhat amusing.

Since the octonions are not associative, however, and since most undergraduate algebra courses deal strictly in objects satisfying the axiom of associativity, it can be difficult to get a handle on them. The purpose of this page is to present the octonions is a fairly transparent manner. This story has been told many times by now, although the approach taken here is a bit novel since we don't use subscripts or the Cayley-Dickson process. We instead present the octonions as an 8-dimensional algebra with 3 generators {a,b,c} subject to some simple algebraic relations. The benefit of not using subscripts to number the basis elements of the octonions is that we are not forced to prefer one of the 480 different meaningful choices, and the benefit of not using the Cayley-Dickson process is that we can focus strictly on the octonions. A disadvantage, perhaps, of the notation presented here is that the expressions for the basis vectors of the octonions may appear a bit long and/or cumbersome.

The first thing to remember about the octonions **O** is that they comprise an algebra.
This means first of all that **O** is a vector space satisfying all the
usual axioms of being a vector space, but also that this vector space also has
an additional operation which for lack of a better term is called "octonion
multiplication". This multiplication operation must further satisfy the distributive
law in order to be considered an algebra. Remembering that the space of
octonions is an algebra is useful because it means that all we need to do
in order to specify the multiplication is to give a basis for the octonions
and then give a multiplication table for products of pairs from this basis.
Notice that we have not imposed the axiom of associativity. For this reason, a vector
space equipped merely with a distributive multiplication rule is also called "non-associative",
even though such an algebra may in fact be associative.

Next, one should also try to remember that the octonions **O** represent a particular extension
of the quaternions **H**, the quaternions represent a particular
extension of the complex numbers **C**, and that the complex numbers represent a particular
extension of the field of real numbers **R**. In each of these extensions, the dimension
of the succeeding algebra (as an algebra over the reals) is twice the dimension of
the preceeding algebra, and the succeeding rule for multiplication is constructed using the preceeding
rule for multiplication. As it turns out, this construction, known as the Cayley-Dickson
process, may be carried out indefinitely, so that one indeed has an infinite family of real
algebras each having dimension 2^{n}. However, we want to focus strictly on
the octonions, so we will not delve into the details of this construction.
For that matter, one should also note that Clifford algebras also provide a way to
generalize complex numbers and quaternions. However, the family of Clifford algebras, although closely related
to the Cayley-Dickson process,
diverge at the point of the octonions because they are all associative algebras.

As the reader should be aware by now, the octonions comprise an 8-dimensional non-associative
algebra which is truly not associative. We are now ready to present a basis for **O**.
It is thus:

The element 1 must commute with all the octonions and serve as an identity element. It is also useful to distinguish between purely real octonions, those which are real multiples of 1, and purely imaginary octonions, those which are linear combinations of the 7 octonions {a,b,ab,c,ac,bc,(ab)c}. Moreover, for some purposes which will arise later, we also postulate that the set B is orthonormal. Thus, we have decomposed

We still need a multiplication table for these basis octonions. Evidently we have already
implied part of this multiplication table by writing down the particular elements of the basis B.
That is,
one can probably guess that

or that

to give a couple examples. One should notice the presence of the parentheses in the expression "(ab)c", and generally remember not to omit parentheses at one's leisure. After all, we have already stated that the octonions are not associative, and it is certainly not true that (ab)c and a(bc) represent the same octonion. Indeed, as we shall see again below, these two octonions are negatives of each other.

Let us now illustrate how to obtain the octonion multiplication table using some of the elementary
properties of the octonions. Squaring octonions is easy. We postulate that 1^{2}=1
and that

To get the squares of the remaining octonions, we must impose a few more rules. First we have a kind of skew-symmetry: The octonion 1 commutes with everything and we also have

Next we have the alternative rule: If x and y are any octonions, then

Notice that the alternative rule is a weak form of associtivity. It guarantees that any subalgebra of the octonions generated by just two octonions must be associative. (In fact, this last statement is equivalent to the alternative rule since our algebra has only 3 generators.)

Now we have a lot to work with. Using the axioms given so far, one can square many of the octonions.
For example, we have

and similarly (ac)

The octonions are postulated to have this property.

We are now ready to write a multiplication table for the octonions. We recall that the pure real octonion
1 serves as an identity and spans the center of **O**, so we don't need to include it in the table.
After working through the axioms presented thus far, one can obtain the following table:

a | b | ab |
c | ac | bc | (ab)c | |

a | -1 | ab | -b | ac | -c | -(ab)c | bc |

b | -ab | -1 | a | bc | (ab)c | -c | -ac |

ab | b | -a | -1 | (ab)c | -bc | ac | -c |

c | -ac | -bc | -(ab)c | -1 | a | b | ab |

ac | c | -(ab)c | bc | -a | -1 | -ab | b |

bc | (ab)c | c | -ac | -b | ab | -1 | -a |

(ab)c | -bc | ac | c | -ab | -b | a | -1 |

This table gives the product xy where x appears in the left-most column and y appears in the top row. For example, using the table, one can see that (ac)(ab)=bc. The skew-symmetry of the octonion product should be clear from the table: Ignoring the diagonal, the left-most column, and the top row, this table represents a skew-symmetric matrix.

**An Action by the Dihedral Group D _{6}**

One can now see an amusing consequence of the octonion axioms given so far,
namely that there are 12 different ways to write the octonion (ab)c using all of the symbols
in {a,b,c} exactly once each. Certainly "(ab)c" is one of these expressions. However,
inside the parentheses of this expression is ab, which by skew-symmetry is identical
to -ba. Thus, we may write (ab)c=-(ba)c, arriving at a second way to write (ab)c.
Using anti-associativity, one may also write (ab)c=-a(bc), giving a third equivalent
expression. In fact, the dihedral group D_{6} of order 12 acts on these expressions. Recall
that D_{6} may be presented by two involutions (also called reflections)
whose composition has order 6. These involutions are already apparent: Let s be the
operation which shifts the parentheses to the left or to the right. Thus the operation
translates the expression (xy)z into -x(yz) and vice-versa, no matter what x, y, and z are.
Next let t be the operation which
switches the order of the factors inside the parentheses. Thus, the operation t translates
(xy)z into -(yx)z and vice-versa. Then one can see that s and t each have order two and that
the product st, acting on these expressions has order 6. Explicitly, one may depict these 12
equivalent expressions with the following diagram:

This diagram clearly exhibits the action of the dihedral group D

It is generally tiresome to build, study, and refer to a multiplication table, and fortunately there are
more geometrically-inspired devices for studying the octonions. In particular,
one may use the Fano plane (also known as the projective plane over the two-element field GF_{2})
as a mnemonic for reproducing the octonion multiplication table. Recall that **O** is
8-dimensional with a center spanned by the octonion 1, so all we really need to know is how
to multiply two pure imaginary quaternions from the set {a,b,ab,c,ac,bc,(ab)c}. Here is one way
to depict this device:

Notice that this diagram has 7 points and through these points are a collection of segments and one circle. These other objects are intended to depict the 3-element lines of the Fano plane. Notice that the bounding triangle has 3 segments with three points on each, 3 segments joining a vertex of the bounding triangle to a point on the opposite side, each with 3 points, and a circle inscribed in the bounding triangle, again passing through exactly 3 points. The variational placement of the lines and circle are basically arbitrary. All we need to do, given these 7 points, is find a way to depict a collection of 7 subsets consisting of 3 points each in such a way that each point lies in exactly 3 of these subsets. Looking at the diagram, one can quickly check that each point lies on exactly 3 of these lines. (One should also verify that this is essentially only one way to do this.)

Clearly the points of the Fano plane are labeled by pure imaginary octonions. The arrows say something about how to multiply them. Notice that the octonions on each line comprise an associative triple. Thus, if the triple {x,y,z} comprises a line, then the set {1,x,y,z} spans an algebra isomorphic to the quaternions, and one has either xy=z or yx=z. If one has xy=z, then, cycling the three symbols, one also has the relations yz=x and zx=y. The same can be said if yx=z, namely that this implies xz=y and zy=x. The arrows are intended to eliminate the ambiguity about whether one has xy=z or yz=x. For example, notice that there is a line consisting of the points {a,bc,(ab)c}. In this case there is an arrow pointing from bc to a, indicating that (bc)a=(ab)c and, again by cycling, that [(ab)c](bc)=a and a[(ab)c]=bc. If one desires to become familiar with the octonions, it is instructive to derive this diagram from scratch and to study what happens to the arrows when some of the octonions are permuted.

**The Exceptional Lie Algebra g _{2}**

The automorphism group of the octonions, as presented here, is a real compact simple 14-dimensional
Lie group known as G_{2}, and, correspondingly, the derivation algebra of the octonions
is a simple 14-dimensional Lie algebra known as g_{2}. (For the octonions, a map
f:**O** --> **O** is an automorphism if f(xy)=f(x)f(y) for all x,y and is a
derivation if f(xy)=xf(y)+f(x)y for all x,y. Looking at the way we have defined
**O**, one should observe the strength of these conditions: They are completely determined
by they way they act on the generators {a,b,c}.) Here we will see how to construct
the Lie algebra g_{2} as a set of derivations on **O**. This construction
conveniently yields the compact form of g_{2} as a Lie subalgebra of the orthogonal Lie algebra
so(7).

The map C

Define another map by

where x and y are octonions, and the meanings of both sets of square brackets are implied above. It takes a little bit of work, but one can show that D

Since the space
of pure imaginary octonions is 7-dimensional and the derivation D_{x,y} is
clearly skew-symmetric in the variables x and y, we see that we appear to have a total
of 21 derivations of **O**. As it turns out, a few of these are redundant. There
are a total of 7 independent relations among all these derivations. One can see these
as follows: Notice that given any element x of our standard basis {a,b,ab,c,ac,bc,(ab)c}
for the imaginary octonions, there are precisely 3 ways to write x as a product of
pairs of the remaining 6. For example, one may write

Each such identity yields a distinct relation. For this one, we get

Thus, the kernel of this map D has dimension at least 7 and the image is a Lie algebra of derivations of dimension at most 14. It takes only a little more work to show that the kernel has dimension exactly 7, so that the dimension of the space of derivations is 14. One still has to check simplicity, although this is not too much work....

**The Hamming Code and Gosset's Lattice**

The Hamming [8,4,4] code H_{8} is distinguished in coding theory as being one
of the simplest examples of a perfect code, meaning that its covering radius
is equal to its packing radius.
It may be constructed using octonions as follows:
First we change the base field from the reals to the two-element field GF_{2}.
Thus, our space of octonions, while still 8-dimensional, has only 2^{8}=256 vectors,
and each of these corresponds naturally to a subset of our basis B={1,a,b,ab,c,ac,bc,(ab)c}.
The Hamming code H_{8} is a linear code, meaning that
it is a vector space, so all we need to do is give a spanning set.

The following describes one way to get a spanning set: First we include the vector

the vector corresponding the entire set B. Next we look again at the Fano plane. Recall that the Fano plane has 7 lines, with elements of B on each line. Thus, for each line, there are four elements, say {w,x,y,z}, which are not on this line. In addition to the vector described above, our spanning set for H

for which the complement of {w,x,y,z} comprise a collinear triple of octonion basis vectors. For example, {a,b,ab} comprise the octonions on one of the seven lines, so we include the octonion

in our spanning set.

Altogether, we have seen a total of 8 vectors in our spanning set for the Hamming code H_{8},
the vector which is the sum of all the octonion basis vectors and another for each of the 7 lines
in the Fano plane.
One can quickly check that exactly four of these are redundant, so one can see that our Hamming code
is a 4-dimensional vector space over the 2-element field. If one defines the weight of a vector
to be the number of terms (or the cardinality of the associated set),
then one can quickly see that there is one vector with weight 0, 14
vectors with weight 4, and 1 vector with weight 8.

Since -1 and 1 represent the same element GF_{2}, our space of octonions using this coefficient field
is in fact associative. Switching back to using the real numbers yields
Gosset's famous 8-dimensional lattice E_{8}, corresponding the densest
packing of spheres in 8-dimensional space. Recall that we have all
those 14 codewords of weight four given above. If you will indulge in some numerology, you will notice
that

and there are precisely 240 roots in the E

is a root vector for E

½(±1 ±a ±c ±ac), ½(±b ±ab ±bc ±(ab)c),

½(±1 ±b ±c ±bc), ½(±a ±ab ±ac ±(ab)c),

½(±1 ±a ±bc ±(ab)c), ½(±b ±ab ±c ±ac),

½(±1 ±b ±ac ±(ab)c), ½(±a ±ab ±c ±bc),

½(±1 ±c ±ab ±(ab)c), ½(±a ±b ±ac ±bc),

½(±1 ±ab ±ac ±bc), ½(±a ±b ±c ±(ab)c).

The remaining 16 are the octonions

So we have described a total of 240 octonions which may or may not have any geometric signifigance
whatsoever. However, one can fit 8 of these octonions (in more than one way)
into the E_{8} Coxeter diagram:

Thus, the 8 octonions appearing in the Coxeter diagram yield 8 reflections generating a group of orthogonal transformations isomorphic to the Coxeter group for this root system. One can check that the orbit of these 8 octonions under this group action is the set of 240 roots given above.

Naturally this is not the only way to construct the E_{8} lattice from the octonions.
The symmetric group on 8 letters obviously acts on the coordinates, and since we found
only 14 codewords with weight 4, we know that we haven't found all of the subspaces of
**O** which are isomorphic to the Hamming Code H_{8}. (Notice that
_{8}C_{4}=70, which is signifigcantly greater than 14.) This leads
to Johannes Kirmse's mistake, a published statement that the lattice generated by
the 240 octonions described above is closed under multiplication. Despite the fact that
it's easy to take a few of these and multiply them together in order to disprove Kirmse's assertion,
it's an easy mistake to make.
After all, this construction presented here has a lot of symmetry, so it seems that there should
be no alternative that they be closed under multiplication. Alas for Kirmse, it is not so.

In spite of Kirmse's mistake, there are some interesting "integral" rings associated with this
lattice. In fact, one can perform an orthogonal transformation on Kirmse's "integers" in such a way
that the image is a non-associative ring. Recall that each octonion is a linear combination of
the octonions {1,a,b,ab,c,ac,bc,(ab)c}. Choosing x to be one of the pure imaginary octonions
in this set, one obtains one of these rings by interchanging 1 and x, and leaving the remaining
6 basis octonions fixed. Evidently the 16 octonions
±x, with x being a basis octonion, always lies in this ring. This interchange operation affects only
the remaining 224 octonions. For example, if we interchange 1 with a, we obtain the
224 octonions:

½(±1 ±a ±c ±ac), ½(±b ±ab ±bc ±(ab)c),

½(±a ±b ±c ±bc), ½(±1 ±ab ±ac ±(ab)c),

½(±1 ±a ±bc ±(ab)c), ½(±b ±ab ±c ±ac),

½(±a ±b ±ac ±(ab)c), ½(±1 ±ab ±c ±bc),

½(±a ±c ±ab ±(ab)c), ½(±1 ±b ±ac ±bc),

½(±a ±ab ±ac ±bc), ½(±1 ±b ±c ±(ab)c).

These 240 octonions serve as the units in a ring whose elements correspond naturally to Gosset's lattice.

It is a remarkable fact that
each of the 7 rings obtained in this way is maximal with respect to a certain integrality condition.
In order to begin to discuss this condition, we need to define the conjugate and norm of an octonion. It goes the
same way as it does for complex numbers and quaternions: If x+y is an octonion, where x is pure real and y is
pure imaginary, then the conjugate is x-y. The norm of the octonion x+y is then defined to be the
real octonion

Every octonion x+y then satisfies a quadratic equation

For each of the 7 rings described above, the coefficients in this equation are all ordinary integers, and they are all maximal with respect to this condition. That is, if one extends any one of them to a larger subring of

**The Magic Square of Freudenthal**

John Baez. The Octonions. *Bulletin of the American Mathematical Society (New Series).*
Volume 39, Number 2, (2002), Pages 145-205.

(Also available online.)

J. H. Conway and N. J. A. Sloane.
*Sphere Packings, Lattices and Groups.* 3rd ed. Springer-Verlag, New York, 1999.

H. S. M. Coxeter. Integral Cayley Numbers. * Duke Mathematical Journal. *
Vol. 13, Number 4, (1946), 561-578.

Richard D. Schafer. *An Introduction to Nonassociative Algebras.*
Dover Publications, Inc., New York, 1994.

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