The Octonions
Yet Another Essay

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 g2, (c) the octonions provide a convenient presentation of Gosset's exceptional 8-dimensional lattice E8 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.

Getting Acquainted 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 2n. 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:

B={1,a,b,ab,c,ac,bc,(ab)c}.

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 O into an orthogonal direct sum of the 1-dimensional space of real octonions and the 7-dimensional space of imaginary octonions. (Hasten to add that the use of the terms "real" and "imaginary" is merely traditional, but otherwise meaningless and arbitrary, serving only to distinguish between two slightly different kinds of objects. All of this stuff is imaginary!)

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

a·b=ab

or that

ab·c=(ab)c

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 12=1 and that

a2=b2=c2=-1.

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

ab+ba=ac+ca=bc+cb=0.

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

x2·y=x·(xy).

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

(ab)2=abab=-a2b2=-1.

and similarly (ac)2=(bc)2=-1. As a matter of fact, we can now say that any subalgebra generated by any two elements of {a,b,c} is isomorphic to the space of quaternions. The next axiom we need is anti-associativity. In order to define this, we must use the orthogonality of the basis given earlier. Notice that any two of the octonions {a,b,c} generates an algebra isomorphic to the quaternions. For example, the algebra generated by a and b is spanned by {1,a,b,ab}, and all four octonions {c,ac,bc,(ab)c} are orthogonal to this subalgebra. This motivates the following definition: We say that a triple T={x,y,z} of octonions comprise an "independent triple" if (a) the algebra generated by any two elements of T is isomorphic to the quaternions, and (b) the algebra generated by any two elements of T is orthogonal to the remaining element of T. For example, {a,b,c} is an independent triple, as are {a,b,bc}, {a,b,c+bc}, {ab,bc,(ab)c}, and so on. Now we can define anti-associativity: For an independent triple {x,y,z}, one always has

x(yz)=-(xy)z.

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

Figure 1. The Octonion Multiplication Table.

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 D6

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 D6 of order 12 acts on these expressions. Recall that D6 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: Figure 2. Expressions Equivalent to (ab)c.

This diagram clearly exhibits the action of the dihedral group D6, with the lines s and t serving as the generating mirrors. As it turns out, the appearance of the dihedral group D6 has signficance. It is the Weyl group for the exceptional 14-dimensional simple Lie algebra g2.

The Fano Plane

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 GF2) 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: Figure 3. The Fano Plane Endowed with Octonions.

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 g2

The automorphism group of the octonions, as presented here, is a real compact simple 14-dimensional Lie group known as G2, and, correspondingly, the derivation algebra of the octonions is a simple 14-dimensional Lie algebra known as g2. (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 g2 as a set of derivations on O. This construction conveniently yields the compact form of g2 as a Lie subalgebra of the orthogonal Lie algebra so(7).

For each octonion x, define a map by

Cx(y)=[x,y]=xy-yx.

The map Cx is not a derivation, but it is an endomorphism of O. Given a pair A, B of endomorphisms, one may unambiguously define their Lie bracket as

[A,B]=AºB-BºA.

Define another map by

Dx,y=[Cx,Cy]+C[x,y],

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 Dx,y is always a derivation of O when x and y are pure imaginary octonions, and that every derivation arises in this way.

Since the space of pure imaginary octonions is 7-dimensional and the derivation Dx,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

ab=a·b=bc·ac=c·(ab)c.

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

Da,b+Dbc,ac+Dc,(ab)c=0.

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 H8 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 GF2. Thus, our space of octonions, while still 8-dimensional, has only 28=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 H8 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

1+a+b+ab+c+ac+bc+(ab)c,

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 H8 also includes those vectors

w+x+y+z

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

c+ac+bc+(ab)c

in our spanning set.

Altogether, we have seen a total of 8 vectors in our spanning set for the Hamming code H8, 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 GF2, 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 E8, 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

16·14+16=240,

and there are precisely 240 roots in the E8 lattice. This is not a coincidence. As it turns out, when we change the field from the 2-element field back to the real numbers, we can use these 14 codewords to generate this lattice. We do this as follows: Suppose w+x+y+z is one of the code words of weight 4 described above. Then we decree that the octonion

½(±w ±x ±y ±z)

is a root vector for E8. Counting all of these, this gives us a total of 16·14 vectors. Explicitly, these are:

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

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

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 E8 Coxeter diagram: Figure 4. The E8 Root System.

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.

Integral Octonions

Naturally this is not the only way to construct the E8 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 H8. (Notice that 8C4=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 ±b ±ab), ½(±c ±ac ±bc ±(ab)c),
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

N(x+y)=(x+y)(x-y).

Every octonion x+y then satisfies a quadratic equation

z2-2xz+N(x+y)=0.

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 O, then the coefficients in this quadratic equation may fail to be integers.

The Magic Square of Freudenthal

I'm sorry, but I still don't understand this completely. Apparently one can use the octonions to build all five of the exceptional complex simple Lie algebras, not just g2. The easiest of the others is probably the construction of the 52-dimensional Lie algebra f4 using an exceptional Jordan algebra. The article by Baez and the book by Schafer also have descriptions of these constructions. (For that matter, one can find most of this stuff elsewhere....)

References

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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