Symmetrically we get (y ∙ x) ∙ x = y ∙ x∙2. □
2.3.1. The Cayley–Dickson doubling process and the generalized Hurwitz theorem
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)
