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Let (, ∙, n) be a Hurwitz algebra, and assume that is a proper unital subalgebra of such that the restriction of n to is non-degenerate. Our goal is to show that in this case also contains a subalgebra obtained by “doubling” , in a way similar to the construction of ℍ from two copies of ℂ, or the construction of from two copies of ℍ.

By non-degeneracy of n, . Pick with n(u) ≠ 0, and let α = −n(u). As , n(u, 1) = 0 and hence and u^∙2 = α1 by the Cayley–Hamilton equation (proposition 2.2). This also implies that , so the right multiplication Ru is bijective.

LEMMA 2.1.– Under the conditions above, the subspaces and are orthogonal (i.e. ) and the following properties hold for any x, :

1 1)

2 2) x ∙ (y ∙ u)= (y ∙ x) ∙ u;

3 3)

4 4) .

PROOF.– For any x, , , so is a subspace orthogonal to .

From , it follows that But , whence .

Now and this is equal, because of [2.6], to .

In a similar vein, , and . □

Therefore, the subspace is also a subalgebra, and the restriction of n to it is non-degenerate. The multiplication and norm are given by (compared to [2.3]):

[2.7]

for any .

Moreover,

while on the other hand

We conclude that , or n(d ∙ (a ∙ c), b) = n((d ∙ a) ∙ c, b).

The non-degeneracy of the restriction of n to implies that is associative. In particular, any proper subalgebra of with non-degenerate restricted norm is associative.

Conversely, given an associative Hurwitz algebra with non-degenerate n, and a non-zero scalar , consider the direct sum of two copies of , with multiplication and norm given by [2.7], extending those on . The arguments above show that (, ∙, n) is again a Hurwitz algebra, which is said to be obtained by the Cayley–Dickson doubling process from (, ∙, n) and α. This algebra is denoted by .

REMARK 2.1.– is associative if and only if is commutative. This follows from x ∙ (y ∙ u) = (y ∙ x) ∙ u. If the algebra is associative, this equals (x ∙ y) ∙ u, and it forces x ∙ y = y ∙ x for any x, . The converse is an easy exercise.

We arrive at the main result of this section.

THEOREM 2.1 (Generalized Hurwitz theorem).– Every Hurwitz algebra over a field is isomorphic to one of the following:

1 1) the ground field ;

2 2) a two-dimensional separable commutative and associative algebra: , with v∙2 = v + μ1, with 4μ +1 ≠ 0, and n(∊ + δv) = ∊2 − μδ2 + ∊δ, for ∊, ;

3 3) a quaternion algebra for as in (2) and ;

4 4) a Cayley (or octonion) algebra , for as in (3) and .

In particular, the dimension of a Hurwitz algebra is restricted to 1, 2, 4 or 8.

PROOF.– The only Hurwitz algebra of dimension 1 is, up to isomorphism, the ground field. If (, ∙, n) is a Hurwitz algebra and , there is an element such that n(v, 1) = 1 and is non-degenerate. The Cayley–Hamilton equation shows that v^∙2 −v + n(v)1= 0, so v^∙2 = v + μ1, with μ = −n(v). The non-degeneracy condition is equivalent to the condition 4μ +1 ≠ 0. Then is a Hurwitz subalgebra of and, if , we are done.

If , we may take an element with n(u) = −β ≠ 0, and hence the subspace is a subalgebra of isomorphic to . By the previous remark, is associative (as is commutative), but it fails to be commutative, as . If , we are done.

Finally, if , we may take an element with n(uʹ) = −γ ≠ 0, and hence the subspace is a subalgebra of isomorphic to , which is not associative by remark 2.1, so it is necessarily the whole . □

Note that if char , the restriction of n to is non-degenerate, so we could have used the same argument for dimension > 1 in the proof above than the one used for > 2. Hence, we get:

COROLLARY 2.1.– Every Hurwitz algebra over a field of characteristic not 2 is isomorphic to one of the following:

1 1) the ground field ;

2 2) a two-dimensional algebra for a non-zero scalar α;

3 3) a quaternion algebra for as in (2) and ;

4 4) a Cayley (or octonion) algebra , for as in (3) and .

REMARK 2.2.– Over the real field ℝ, the scalars α, β and γ in corollary 2.1 can be taken to be ±1. Note that [2.3] and the analogous equation for ℂ and give isomorphisms , and .

REMARK 2.3.– Hurwitz (1898) only considered the real case with a positive definite norm. Over the years, this was extended in several ways. The actual version of the generalized Hurwitz theorem seems to appear for the first time in Jacobson (1958) (if char ) and van der Blij and Springer (1959).

The problem of isomorphism between Hurwitz algebras of the same dimension relies on the norms:

PROPOSITION 2.3.– Two Hurwitz algebras over a field are isomorphic if and only if their norms are isometric.

PROOF.– Any isomorphism of Hurwitz algebras is, in particular, an isometry of the corresponding norms, due to the Cayley–Hamilton equation. The converse follows from Witt’s cancellation theorem (see Elman et al. (2008, theorem 8.4)). □

A natural question is whether the restriction of the dimension of a Hurwitz algebra to be 1, 2, 4 or 8 is still valid for arbitrary composition algebras. The answer is that this is the case for finite-dimensional composition algebras.

COROLLARY 2.2.– Let (, ∙, n) be a finite-dimensional composition algebra. Then its dimension is either 1, 2, 4 or 8.

PROOF.– Let be an element of non-zero norm. Then satisfies n(u) = 1. Using the so-called Kaplansky’s trick (Kaplansky 1953), consider the new multiplication

Note that since the left and right multiplications by a norm 1 element are isometries, we still have n(x ◊ y) = n(x)n(y), so (, ◊, n) is a composition algebra too. But for any x, so the element u^∙2 is the unity of (, ◊) and (, ◊, n) is a Hurwitz algebra, and hence is restricted to 1, 2, 4. or 8. □

However, contrary to the thoughts expressed in Kaplansky (1953), there are examples of infinite-dimensional composition algebras. For example (see Urbanik and Wright (1960)), let φ : ℕ × ℕ → ℕ be a bijection (for instance, φ(n, m) = 2ⁿ⁻¹ (2m − 1)), and let be a vector space over a field of characteristic not 2 with a countable basis {un : n ∈ ℕ}. Define a multiplication and a norm on by

Then (, ∙, n) is a composition algebra.

In Elduque and Pérez (1997), one may find examples of infinite-dimensional composition algebras of arbitrary infinite dimension, which are even left unital.