nLab composition algebra


This entry is about algebras exhibiting “composition of sums of squares”. For the un-related notion of rings exhibiting a structure akin to composition of endofunctions see at composition ring.



Let kk be a field with characteristic char(k)2char(k) \neq 2. A unital composition algebra over kk consists of a finite-dimensional vector space VV together with a

  • a nondegenerate symmetric bilinear form

    ,:VVk, \langle - , - \rangle: V \otimes V \to k \,,
  • a multiplication map, i.e., a bilinear map

    :VVV, - \cdot - \;\colon\; V \otimes V \to V \,,
  • a unit eVe \in V for the multiplication, i.e., so that ev=v=vee \cdot v = v = v \cdot e,

such that:

uv,uv=u,uv,v \langle u\cdot v ,\, u \cdot v \rangle \;=\; \langle u,u \rangle \langle v,v \rangle

The bilinear form defines a norm N(u)=u,uN(u) = \langle u,u\rangle.

Beware that there are no assumptions on the multiplication such as associativity, commutativity, etc.

Examples of composition algebras include the real numbers, the complex numbers, the quaternions, the octonions, and the algebra of 2 × 2 2 \times 2 matrices over a field.

Since char(k)2char(k) \neq 2, we can recover the bilinear form from the norm by the formula

u,v=N(u+v)N(u)N(v)2\langle u, v \rangle = \frac{N(u + v) - N(u) - N(v)}{2}

Since the bilinear form is nondegenerate, we may infer u=vu = v whenever

u,w=v,w\langle u, w \rangle = \langle v, w \rangle

and this will be frequently used in the sequel.

Also since the form is nondegenerate, there exists vVv \in V such that N(v)0N(v) \neq 0. From N(v)=N(ev)=N(e)N(v)N(v) = N(e v) = N(e)N(v), it follows that N(e)=1N(e) = 1.

Para- and non-unital composition algebras

Frequently, one refers to unital composition algebras simply as composition algebras. However, this unital property can be weakened or removed altogether. A weakening of this condition leads to para-unital composition algebras. This requires the existence of an involution

xx¯ x\mapsto\bar{x}

such that there exists a distinguished element 1, called the para-unit, such that

x1=1x=x¯ x\cdot \mathbf{1} = \mathbf{1}\cdot x = \bar{x}

meaning 1 acts as a unit up to an involution.

On the other hand, removing the existence of a unit altogether leads to the notion of non-unital composition algebras.

These two generalizations of composition algebras are intimately related to a generalization of the Hurwitz theorem (see generalized Hurwitz theorem for the statement).


Basic identities

The arrangements of the proofs below are based in part on the treatments by Conway and Smith, and by Springer and Veldkamp (see references below).


(Scaling) uv,uw=N(u)v,w\langle u v, u w \rangle = N(u)\langle v, w \rangle and uw,vw=u,vN(w)\langle u w, v w \rangle = \langle u, v \rangle N(w)


The left sides, and therefore the right sides of the equations below are equal:

N(u(v+w))=N(u)N(v+w)=N(u)(N(v)+2v,w+N(w))N(u(v + w)) = N(u)N(v + w) = N(u)(N(v) + 2\langle v, w \rangle + N(w))
N(uv+uw)=N(uv)+2uv,uw+N(uw)=N(u)N(v)+2uv,uw+N(u)N(w)N(u v + u w) = N(u v) + 2\langle u v, u w \rangle + N(u w) = N(u)N(v) + 2\langle u v, u w \rangle + N(u)N(w)

and the result follows by cancellation and division by 22.


(Exchange) uv,wx=2u,wv,xux,wv\langle u v, w x \rangle = 2\langle u, w \rangle \langle v, x \rangle - \langle u x, w v \rangle


From the scaling identity, we have

(u+w)v,(u+w)x=N(u+w)v,x=N(u)v,x+2u,wv,x+N(w)v,x=uv,ux+2u,wv,x+wv,wx\langle (u + w)v, (u + w)x \rangle = N(u + w)\langle v, x \rangle = N(u)\langle v, x \rangle + 2\langle u, w \rangle \langle v, x \rangle + N(w)\langle v, x \rangle = \langle u v, u x \rangle + 2\langle u, w \rangle \langle v, x \rangle + \langle w v, w x \rangle

but the left-hand side is equal to

uv+wv,ux+wx=uv,ux+wv,ux+uv,wx+wv,wx\langle u v + w v, u x + w x \rangle = \langle u v, u x \rangle + \langle w v, u x \rangle + \langle u v, w x \rangle + \langle w v, w x \rangle

and now we equate the right-hand sides and cancel to get the result.

Conjugation identities

In any composition algebra, we may define a conjugation operator by

v¯=2v,eev\bar{v} = 2\langle v, e \rangle e - v

Observe that v¯=v\bar{v} = v just when vv is a scalar multiple of the identity. By analogy with the classical case (composition algebras over \mathbb{R}), such elements will be called real.

The next few propositions develop properties of conjugation.


(Adjointness) uv,w=v,u¯w\langle u v, w \rangle = \langle v, \bar{u}w \rangle and uv,w=u,wv¯\langle u v, w \rangle = \langle u, w\bar{v} \rangle. w,uv=u¯w,v\langle w, u v \rangle = \langle \bar{u} w, v \rangle and w,uv=wv¯.u\langle w, u v \rangle = \langle w\bar{v}. u \rangle.


Put x=ex = e in the exchange identity to get the first equation in

uv,w=2u,wv,eu,wv=u,w(2v,eev)=u,wv¯\langle u v, w \rangle = 2\langle u, w \rangle \langle v, e \rangle - \langle u, w v \rangle = \langle u, w(2\langle v, e\rangle e - v)\rangle = \langle u, w\bar{v} \rangle

The second adjointness equation is proved similarly; the final two come from symmetry of the form.


(Involution) v=v¯¯v = \bar{\bar{v}} for all vv.


For all uu we have

u,v=uv¯,e=u,v¯¯\langle u, v \rangle = \langle u\bar{v}, e \rangle = \langle u, \bar{\bar{v}} \rangle

and the result follows from nondegeneracy.


(Unitarity) u,v=v¯,u¯=u¯,v¯\langle u, v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle.


u,v=e,u¯v=v¯,u¯=u¯,v¯\langle u, v \rangle = \langle e, \bar{u}v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle where the last equation is symmetry of the bilinear form.


(Anti-automorphism) u¯v¯=vu¯\bar{u} \bar{v} = \widebar{v u}.


For all ww we have

u¯v¯,w=v¯,uw=v¯w¯,u=w¯,vu=vu¯,w\langle \bar{u}\bar{v}, w \rangle = \langle \bar{v}, u w \rangle = \langle \bar{v}\bar{w}, u \rangle = \langle \bar{w}, v u \rangle = \langle \widebar{v u}, w \rangle

using involution and unitarity. The result follows from nondegeneracy of the form.

By the involution and anti-automorphism properties, we see that v¯v\bar{v}v is fixed under conjugation: is “real”. Better yet,


(Reality) u¯(uv)=N(u)v\bar{u} \cdot (u v) = N(u)v.


For all ww,

u¯(uv),w=uv,uw=N(u)v,w=N(u)v,w\langle \bar{u}\cdot (u v), w \rangle = \langle u v, u w \rangle = N(u)\langle v, w \rangle = \langle N(u)v, w \rangle

and the result follows from nondegeneracy.

This last result has several interesting corollaries. Putting v=ev = e, we see that

  • N(u)0N(u) \neq 0 implies uu is invertible, with u 1=u¯/N(u)u^{-1} = \bar{u}/N(u).

  • N(u)=0N(u) = 0 implies uu is a zero divisor, with u¯u=0\bar{u} u = 0.

In either case, we have from u¯=2u,eeu\bar{u} = 2\langle u, e \rangle e - u the identity

u¯u=(2u,eeu)u=N(u)e\bar{u} u = (2\langle u, e \rangle e - u)u = N(u)e

so that every element uu of a composition algebra satisfies a quadratic equation

u 22u,eu+N(u)e=0.u^2 - 2\langle u, e \rangle u + N(u)e = 0.

This has as further consequence the fact that an algebra admits at most one norm making it a composition algebra (because the minimal monic polynomial of an element uu in a finite-dimensional algebra is uniquely determined; the norm of an element would the uniquely determined constant coefficient of its minimal polynomial).

A final corollary of Reality is


(Alternative law) u(uv)=u 2vu \cdot (u v) = u^2 \cdot v and uv 2=(uv)vu \cdot v^2 = (u v) \cdot v.


We have w(uv)=(wu)vw(u v) = (w u)v if ww is either ee or u¯\bar{u}, and uu is a linear combination of ee and u¯\bar{u}. The other equation is proven similarly.

These are the two axioms as given in alternative algebra, but we remark that often a third alternative law is considered: u(vu)=(uv)uu (v u) = (u v) u. For discussion of this in composition algebras, see the section on Moufang identities below.

Cayley-Dickson doubling construction

This is essentially the same as the Cayley-Dickson construction, but in this section it is applied specifically to composition algebras where we have to deal with a norm, whereas the general construction applies to general (nonassociative) algebras equipped with an anti-involution.

We begin with a simple observation:


Let VV be a finite-dimensional vector space with a nondegenerate bilinear form, and let WW be a subspace such that the form on VV restricts to a nondegenerate form on WW. Then

V=WW V = W \oplus W^\perp

and the form on VV restricts to a nondegenerate form on W W^\perp.


The fact that WW ={0}W \cap W^\perp = \{0\} is immediate from nondegeneracy of the form on WW, and that W+W =VW + W^\perp = V follows from this and the fact that dim(W)+dim(W )=dim(V)dim(W) + dim(W^\perp) = dim(V) (use dim(W )=dim((V/W) *)=dim(V/W)dim(W^\perp) = dim((V/W)^*) = dim(V/W) and dim(V)=dim(W)+dim(V/W)dim(V) = dim(W) + dim(V/W)). For the second assertion, we know that for vW v \in W^\perp, the map v,| W:Wk\langle v, - \rangle |_W: W \to k is zero; if also v,| W :W k\langle v, - \rangle |_{W^\perp}: W^\perp \to k is zero, then v,:Vk\langle v, - \rangle: V \to k is zero because V=W+W V = W + W^\perp, and v=0v = 0 follows from nondegeneracy of the form on VV.

Thus, given a composition algebra VV and a composition subalgebra WW of VV (that is, a subspace closed under identity and multiplication, such that the norm on VV restricts to a nondegenerate form on WW), the proposition shows there exists αW \alpha \in W^\perp such that N(α)0N(\alpha) \neq 0. This α\alpha is invertible, so αW\alpha \cdot W has the same dimension as WW. Moreover, for all v,wWv, w \in W we have

αv,w=α,wv¯=0\langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0

so that, by nondegeneracy of the form on WW, αWW={0}\alpha W \cap W = \{0\}. Indeed, αW\alpha W is orthogonal to WW. It follows that W+αWW + \alpha W has double the dimension of WW.

Now let us fix such an α\alpha, and put λ=N(α)\lambda = N(\alpha).


For elements u,v,w,xWu, v, w, x \in W,

u+αv,w+αx=u,w+λv,x.\langle u + \alpha v, w + \alpha x \rangle = \langle u, w \rangle + \lambda \langle v, x \rangle.

This follows from the equations

u,αx=ux¯,α=0αv,w=α,wv¯=0αv,αx=N(α)v,x\langle u, \alpha x \rangle = \langle u \bar{x}, \alpha \rangle = 0 \qquad \langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0 \qquad \langle \alpha v, \alpha x \rangle = N(\alpha)\langle v, x \rangle

plus bilinearity of the form.

Consequently, if u+αv,w=0\langle u + \alpha v, w \rangle = 0 for all wWw \in W, we must have u=0u = 0, and if u+αv,αx=0\langle u + \alpha v, \alpha x \rangle = 0 for all xWx \in W, then v=0v = 0. It follows that the form on VV, when restricted to W+αWW + \alpha W, is nondegenerate.

Now we want to show that the double W+αWW + \alpha W is closed under multiplication, hence forms a composition subalgebra. It follows immediately from all this that, starting from the trivial composition subalgebra kek \cdot e of dimension 1, dim(V)dim(V) must be a power of 2, and in fact we will see later that the only possible dimensions are 1, 2, 4, and 8. Indeed, the possible structures of composition algebras are very tightly constrained.


(Conjugation on the double) We have u+αv¯=u¯αv\widebar{u + \alpha v} = \bar{u} - \alpha v. Consequently, αv=αv¯=v¯α¯=v¯α\alpha v = - \widebar{\alpha v} = - \bar{v} \bar{\alpha} = \bar{v} \alpha, and α¯=α\widebar{\alpha} = -\alpha.


αv¯=2αv,eeαv=αv\widebar{\alpha v} = 2\langle \alpha v, e \rangle e - \alpha v = -\alpha v.


(Closure under multiplication) For all u,v,w,xWu, v, w, x \in W, (u+αv)(w+αx)=(uwλxv¯)+α(wv+u¯x)(u + \alpha v)(w + \alpha x) = (u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x).


For all yVy \in V, we have the following sets of equations, using the previous proposition (Conj), the Exchange identity (Ex), and other identities frequently observed above (unlabeled as such).

u(αx),y=αx,u¯y=Ex0αy,u¯x=y,α(u¯x)=α(u¯x),y\langle u (\alpha x), y \rangle = \langle \alpha x, \bar{u} y \rangle \stackrel{Ex}{=} 0 - \langle \alpha y, \bar{u} x \rangle = \langle y, \alpha (\bar{u} x) \rangle = \langle \alpha (\bar{u} x), y \rangle
(αv)w,y=αv,yw¯=Conjv¯α,yw¯=Ex0v¯w¯,yα=(v¯w¯)α,y=wv¯α,y=Conjα(wv),y\langle (\alpha v)w, y \rangle = \langle \alpha v, y \bar{w} \rangle \stackrel{Conj}{=} \langle \bar{v} \alpha, y \bar{w} \rangle \stackrel{Ex}{=} 0 - \langle \bar{v}\bar{w}, y \alpha \rangle = \langle (\bar{v}\bar{w})\alpha, y \rangle = \langle \widebar{w v} \alpha, y \rangle \stackrel{Conj}{=} \langle \alpha (w v), y \rangle
(αv)(αx),y=Conjαv,y(αx)=Ex0+α(αx),yv=Conjαx,α(yv)=λx,yv=λxv¯,y\langle (\alpha v)(\alpha x), y \rangle \stackrel{Conj}{=} -\langle \alpha v, y (\alpha x) \rangle \stackrel{Ex}{=} 0 + \langle \alpha (\alpha x), y v \rangle \stackrel{Conj}{=} -\langle \alpha x, \alpha (y v) \rangle = -\lambda \langle x, y v \rangle = \langle -\lambda x\bar{v}, y \rangle

These identities, combined with nondegeneracy of the form, give the result.

Possible dimensions are 1, 2, 4, and 8.

The calculation expressed by the fundamental theorem just stated has some remarkable consequences:

  • Suppose V=W+αWV = W + \alpha W. Then WW is an associative composition algebra.

For, by starting from the identity

N(u+αv)N(w+αx)=N((uwλxv¯)+α(wv+u¯x))N(u + \alpha v)N(w + \alpha x) = N((u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x))

and expanding, one obtains

(N(u)+λN(v))(N(w)+λN(x))=N(uw)2λuw,xv¯+λ 2N(xv¯)+λ(N(wv)+2wv,u¯x+N(u¯x))(N(u) + \lambda N(v))(N(w) + \lambda N(x)) = N(u w) - 2\lambda \langle u w, x\bar{v} \rangle + \lambda^2 N(x\bar{v}) + \lambda (N(w v) + 2\langle w v, \bar{u} x \rangle + N(\bar{u}x))

Using the fact that NN is a homomorphism, plus unitarity N(u)=N(u¯)N(u) = N(\bar{u}), further expansions and cancellations yield

0=2λuw,xv¯+2λwv,u¯x0 = -2\lambda \langle u w, x\bar{v} \rangle + 2\lambda \langle w v, \bar{u}x \rangle

which, by adjointness, yields

(uw)v,x=u(wv),x\langle (u w)v, x \rangle = \langle u(w v), x \rangle

which by nondegeneracy on WW, yields associativity (uw)v=u(wv)(u w)v = u(w v).

  • Suppose V=W+αWV = W + \alpha W is an associative composition algebra. Then WW is a commutative, associative composition algebra.

For clearly the subalgebra WW must be associative; it is also commutative via the following string of equations (using conjugation of the double):

α(vw)=(αv)w=(v¯α)w=v¯(αw)=v¯(w¯α)=(v¯w¯)α=wv¯α=α(wv)\alpha(v w) = (\alpha v)w = (\bar{v}\alpha)w = \bar{v}(\alpha w) = \bar{v}(\bar{w} \alpha) = (\bar{v}\bar{w})\alpha = \widebar{w v}\alpha = \alpha (w v)

and cancelling out α\alpha.

Conversely, a lengthy but straightforward calculation shows that if WW is commutative and associative, then VV is associative.

  • Suppose V=W+αWV = W + \alpha W is a commutative associative composition algebra. Then WW is purely real, i.e., is the trivial 1-dimensional associative commutative algebra kek \cdot e.

This results from

αw=Conjw¯α=commαw¯\alpha w \stackrel{Conj}{=} \bar{w}\alpha \stackrel{comm}{=} \alpha \bar{w}

so that w=w¯w = \bar{w} for every wWw \in W, so that ww is real. Conversely, from

(u+αv)(w+αx)=(uwλxv¯)+α(wv+u¯x)(u + \alpha v)(w + \alpha x) = (u w - \lambda x\bar{v}) + \alpha (w v + \bar{u} x)
(w+αx)(u+αv)=(wuλvx¯)+α(ux+w¯v)(w + \alpha x)(u + \alpha v) = (w u - \lambda v\bar{x}) + \alpha (u x + \bar{w} v)

together with commutativity and trivial conjugation in WW, we infer commutativity in VV.

Hence the doubling process may be iterated three times at most.

This same result can also be proven using string diagram calculus. See this paper for a nice exposition of that route.

Hurwitz’s Theorem

The classification of composition algebras over specific fields (e.g., number fields, local fields) can be a bit intricate; in this section we concentrate solely on the classical case where k=k = \mathbb{R} is the real numbers, where the results have been known for a long time, known as the Hurwitz theorem.

A fundamental dichotomy is whether or not the composition algebra has zero divisors, i.e., elements vv such that N(v)=0N(v) = 0. If not, then the composition algebra is a division algebra (every nonzero element is invertible). If so, then the composition algebra is called a split composition algebra. We analyze each in turn.


In a division composition algebra, all nonzero elements have positive norm.


If all elements vv orthogonal to the identity ee have positive norm, the result is immediate since

N(rv+se)=r 2N(v)+s 20N(r v + s e) = r^2 N(v) + s^2 \geq 0

Otherwise, if some such element vv has N(v)=λ<0N(v) = \lambda \lt 0, we may put u=v/|λ| 1/2u = v/|\lambda|^{1/2} so that N(u)=1N(u) = -1. Then uu is orthogonal to ee and

N(u+e)=N(u)+N(e)=1+1=0N(u + e) = N(u) + N(e) = -1 + 1 = 0

which contradicts the assumption that all nonzero elements are invertible.

In particular, any division composition algebra is a normed division algebra.

Now let VV be a division composition algebra, with V=W+αWV = W + \alpha W, where 0αW 0 \neq \alpha \in W^\perp. Put j=α/N(α) 1/2j = \alpha/N(\alpha)^{1/2}, so that N(j)=1N(j) = 1, jWj \perp W, and V=W+jWV = W + j W. We have the following possibilities.

  • dim(V)=2dim(V) = 2. In that case WW is purely real and VV is a commutative field over \mathbb{R} with j 2=jj¯=N(j)=1-j^2 = j\bar{j} = N(j) = 1. This is of course the complex numbers, with

    N(s+jt)=s 2+t 2N(s + j t) = s^2 + t^2

    the usual norm. The conjugate of s+jts + j t is sjts - j t.

  • dim(V)=4dim(V) = 4. In that case WW is a 2-dimensional division composition algebra, hence isomorphic to \mathbb{C}, and VV is an associative division algebra over \mathbb{R} given by V=+jV = \mathbb{C} + j\mathbb{C}, where again j 2=1j^2 = -1. (Evidently VV is not commutative because WW is not purely real.) By conjugation of the double, we have

    ji=ijj i = -i j

    where ii is an imaginary unit of \mathbb{C}, and we arrive at the algebra of quaternions \mathbb{H} over \mathbb{R}, with orthonormal basis provided by 1,i,j,k=ij1, i, j, k = i j. Conjugation is given by the usual operation

    a+bi+cj+dkabicjdka + b i + c j + d k \mapsto a - b i - c j - d k
  • dim(V)=8dim(V) = 8. In that case WW is a 4-dimensional division composition algebra, hence isomorphic to \mathbb{H}, and VV is an alternative division algebra over \mathbb{R} given by V=+jV = \mathbb{H} + j\mathbb{H}, with j 2=1j^2 = -1. (VV is not associative because WW is not commutative.) The structure of multiplication is given by the theorem above and the resulting algebra is the algebra of octonions, with the standard norm and conjugation.

Thus, we have established the Hurwitz theorem


(Hurwitz) The only division composition algebras over the real numbers \mathbb{R} are the real numbers, complex numbers, quaternions, and octonions.

Split composition algebras

Now we turn to split composition algebras VV. It turns out that the structure of these is not specific to the field \mathbb{R}: the classification of possible split composition algebras is the same over any field (see the text by Springer and Veldkamp), although we will continue to work over \mathbb{R} as we describe them below.

Suppose V=W+αWV = W + \alpha W, where αW \alpha \in W^\perp, N(α)0N(\alpha) \neq 0. Put j=α/|N(α)| 1/2j = \alpha/|N(\alpha)|^{1/2}, so |N(j)|=1|N(j)| = 1, V=W+jWV = W + j W. In addition to the trivial 1-dimensional case, we have the following possibilities.

  • dim(V)=2dim(V) = 2. In this case N(j)=1N(j) = -1 (else VV would be a division algebra, not a split composition algebra) and j 2=1j^2 = 1 (we are now using 11 to denote the identity). The elements

    e 1=1+j2e 2=1j2e_1 = \frac{1 + j}{2} \qquad e_2 = \frac{1-j}{2}

    are primitive idempotents, conjugate to one another, and Ve 1e 2V \cong \mathbb{R} e_1 \oplus \mathbb{R} e_2 as a product ring. The norm of an element xe 1+ye 2x e_1 + y e_2 is N(xe 1+ye 2)=xyN(x e_1 + y e_2) = x y.

  • dim(V)=4dim(V) = 4. Let ii be an imaginary unit of WW, so i¯=i\bar{i} = -i and |N(i)|=1|N(i)| = 1. Here either N(i)=1N(i) = -1 (WW is split), or N(i)=1N(i) = 1 (WW is isomorphic to \mathbb{C}). In the second instance, N(j)=1N(j) = -1, else VV would be a division algebra, and we may replace WW by the split algebra W=+ijW' = \mathbb{R} + \mathbb{R} i j and still have V=W+jWV = W' + j W'. So without loss of generality we may assume WW is split; therefore, there is up to isomorphism only one split composition algebra of dimension 4. This is the algebra of 2×22 \times 2 matrices AA, for which N(A)=det(A)N(A) = det(A) and WW is embedded as the subalgebra of diagonal matrices; the element jj may be taken to be the matrix AA with a 11=a 22=0a_{11} = a_{22} = 0, a 12=a 21=1a_{12} = a_{21} = 1. The conjugate of a matrix AA is A¯=Tr(A)IA\bar{A} = Tr(A)I - A, which leads to the familiar formula for det(A)A 1det(A) A^{-1} when AA is invertible.

  • dim(V)=8dim(V) = 8. Again, by an argument similar to the one used for the case of dimension 4, we may assume a maximal proper composition subalgebra WW is split, and up to isomorphism there is only one split composition algebra of dimension 8, aka the split octonions. The multiplication may be deduced from the fundamental result on doubling multiplication above, or may be expressed as follows. Denote scalars by letters like r,sr, s and 3-vectors by letters like x,yx, y. Let x,y\langle x, y \rangle denote the standard inner product

    x 1y 1+x 2y 2+x 3y 3x_1 y_1 + x_2 y_2 + x_3 y_3

    and let xyx \wedge y denote the standard cross-product, so that xy,z=det(x,y,z)\langle x \wedge y, z \rangle = det(x, y, z). Elements of VV are represented by 2×22 \times 2 arrays

    (r x y s)\left( \begin{aligned} r & x\\ y & s \end{aligned} \right)

    and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:

    (r x y s)(r x y s)=(rr+x,y rx+sx+yy ry+sy+xx y,x+ss)\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) \cdot \left( \begin{aligned} r' & x'\\ y' & s' \end{aligned} \right) = \left( \begin{aligned} r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\ r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s' \end{aligned} \right)

    The norm is given by a kind of determinant formula

    N(r x y s)=rsx,yN\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) = r s - \langle x, y \rangle

Moufang identities

Further consequences of the composition algebra axioms include the Moufang laws which are important in the study of octonions.

Moufang identities

  • (uv)(wu)=(u(vw))u)=u((vw)u)(u v)(w u) = (u(v w))u) = u((v w)u)

  • ((uv)u)w=u(v(uw))((u v)u)w = u(v(u w))

  • ((uv)w)v=u(v(wv))((u v)w)v = u(v(w v))

We will prove the first of these; the others are proven in similar style (see Springer-Veldkamp for details). (It may be tricky to remember how the bracketings go, but one thing to remember is that the bracketings shouldn’t lead to proofs of general associativity when interpreted in a division algebra!)


We have

(uv)(wu),x = uv,x(u¯w¯) =Ex 2u,xv,u¯w¯u(u¯w¯),xv = 2u,xvw,u¯u¯w¯,u¯(xv) = 2vw,u¯u,xN(u)w¯v¯,x = 2vw,u¯u,xN(u)vw¯,x\array{ \langle (u v)(w u), x \rangle & = & \langle u v, x(\bar{u}\bar{w})\rangle\\ & \stackrel{Ex}{=} & 2\langle u, x\rangle\langle v, \bar{u}\bar{w} \rangle - \langle u(\bar{u}\bar{w}), x v \rangle \\ & = & 2\langle u, x\rangle\langle v w, \bar{u} \rangle - \langle \bar{u}\bar{w}, \bar{u}(x v) \rangle \\ & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \bar{w}\bar{v}, x \rangle \\ & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \widebar{v w}, x \rangle }

which makes it plain that (uv)(wu)(u v)(w u) depends on uu and vwv w only. Hence we get the same result if we replace vv and ww and any two elements whose product is vwv w, say vwv w and ee. In other words,

(uv)(wu)=(u(vw))(eu)=(u(vw))u,(uv)(wu)=(ue)((vw)u)=u((vw)u)(u v)(w u) = (u(v w))(e u) = (u(v w))u, \qquad (u v)(w u) = (u e)((v w)u) = u((v w)u)

which completes the proof.


For all uu, vv in a composition algebra, the third alternative law holds: u(vu)=(uv)uu(v u) = (u v)u.

See also Moufang loop.


From Ab to monoidal categories

This concept could be generalized from the category of abelian groups to any monoidal category, since kk-vector spaces are kk-modules when kk is a field:

Let (C,I,)(C, I, \otimes) be a monoidal category, let (k,1,π k)(k, 1, \pi_k) be a monoid object in CC. kk itself is a kk-module object with the action being represented by the monoid binary operation π k\pi_k. A (nonunital) composition algebra object in CC is a kk-module object (A,ρ)(A, \rho) with

  • an action-preserving morphism f:VVkf:V \otimes V \to k from the tensor product VVV \otimes V to kk: given morphisms a:Ika:I \to k, v:IVv:I \to V, and w:IVw:I \to V,
f((ρ(av))w)=π(a(f(vw))f \circ ((\rho \circ (a \otimes v)) \otimes w) = \pi \circ (a \otimes (f \circ (v \otimes w))


f(v(ρ(aw)))=π(a(f(vw))f \circ (v \otimes (\rho \circ (a \otimes w))) = \pi \circ (a \otimes (f \circ (v \otimes w))
  • a morphism π A:AAA\pi_A \colon A \otimes A \to A

  • such that for all morphisms u:IAu \colon I \to A and v:IAv \colon I \to A,

(1)f(π A(uu))(π A(vv))=π k((f(uv))(f(uv))). f \circ \Big( \pi_A \circ (u \otimes u) \big) \otimes \big( \pi_A \circ (v \otimes v) \Big) \;\;=\;\; \pi_k \circ \Big( \big( f \circ (u \otimes v) \big) \otimes \big( f \circ (u \otimes v) \big) \Big) \,.

A paraunital composition algebra object in CC is a composition algebra object AA which is also a paraunital algebra object with respect to the morphism π A:AAA\pi_A:A \otimes A \to A, and a unital composition algebra object in CC is a composition algebra object AA which is also a unital algebra object with respect to the morphism π A:AAA\pi_A:A \otimes A \to A.

From Vect to monoidal categories

This concept may be generalized from the category of vector spaces to any monoidal category, observing that the ground field kk is the tensor unit of the category of kk-vector spaces:

Let (C,I,)(C, I, \otimes) be a monoidal category, where II is a monoid object with unit e:IIe:I \to I and binary operation π I:III\pi_{I}:I \otimes I \to I. Then a (nonunital) composition algebra object in CC is an object ACA \in C with a morphism π A:AAA\pi_A \colon A \otimes A \to A and f:AAIf \colon A \otimes A \to I such that for all morphisms u:IAu:I \to A and v:IAv \colon I \to A,

(2)f((π A(uu))(π A(vv)))=π I((f(uv))(f(uv))). f \circ \Big( \big( \pi_A \circ (u \otimes u) \big) \otimes \big( \pi_A \circ (v \otimes v) \big) \Big) \;\;=\;\; \pi_{I} \circ \Big( \big( f \circ (u \otimes v) \big) \otimes \big( f \circ (u \otimes v) \big) \Big) \,.

A paraunital composition algebra object in CC is a composition algebra object AA which is also a paraunital algebra object with respect to the morphism π A:AAA\pi_A:A \otimes A \to A, and a unital composition algebra object in CC is a composition algebra object AA which is also a unital algebra object with respect to the morphism π A:AAA\pi_A:A \otimes A \to A.

In cartesian monoidal categories (C,1,×)(C, 1, \times), since any morphism V×V1V \times V \to 1 exists and is unique by the universal property of the terminal object, the only monoid structure on 11 is the trivial monoid structure given by the identity function on 11 and the (left or right) unitor on 11 respectively, and composition algebra objects are the same as magma objects.


The term “composition algebra” refers to “composition of sums of squares”, as in

  • Olga Taussky, Section 6 of: Sums of squares, The American Mathematical Monthly 77 8 (1970) 805-830 [[doi:10.2307/2317016]]

Textbook account:


A general abstract formulation of Rost 96

  • Markus Rost, On the dimension of a composition algebra, Documenta Mathematica 1 (1996), 209-214, files Abstract: “The possible dimensions of a composition algebra are 1, 2, 4, or 8. We give a tensor categorical argument.”

in terms of string diagrams in additive braided monoidal categories is in

An exposition of the string diagram proof of the Hurwitz’ theorem on the classification of compositon algebras is given in

See also

  • John Conway, Derek A. Smith, On Quaternions and Octonions, A.K. Peters, 2003.

  • T.A. Springer, F.D. Veldkamp, Octonions, Jordan algebras, and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag 2000.

  • wikipedia

  • eom: Lie-admissible algebra

A proposed application of (non-unital) composition algebras in QCD is described in

Last revised on October 31, 2023 at 19:04:59. See the history of this page for a list of all contributions to it.