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 .
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]):
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 .
also contains a subalgebra obtained by “doubling” 