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Assume now that the norm of a Hurwitz algebra (, ∙, n) represents 0. That is, there is a non-zero element such that n(a) = 0. This is always the case if and is algebraically closed.

With a as above, take such that , so that n(a ∙ b, 1) = 1. Also n(a ∙ b) = n(a)n(b) = 0. By the Cayley–Hamilton equation, the non-zero element e₁ := a ∙ b satisfies , that is, e₁ is an idempotent. Consider too the idempotent , and the subalgebra generated by e₁. (1= e₁ + e₂).

For any and, as and , we conclude that x ∙ e₁ = e₂ ∙ x, and in the same way, x ∙ e₂ = e₁ ∙ x, for any .

But x = 1 ∙ x = e₁ ∙ x + e₂ ∙ x, and e₂ ∙ (e₁ ∙ x)= (1 − e₁) ∙ (e₁ ∙ x) = 0 = e₁ ∙ (e₂ ∙ x). It follows that splits as with

For any , n(u) = n(e₁ ∙ u) = n(e₁₎n(u) = 0, so , and too, are totally isotropic subspaces of paired by the norm. In particular, , and this common value is either 0, 1 or 3, depending on being 2, 4 or 8. The case of = 0 is trivial, and the case of = 1 is quite easy (and subsumed in the arguments below). Hence, let us assume that is a Cayley algebra (dimension 8), so .

For any and , using [2.5] we get

Hence, is orthogonal to both and , so it must be contained in . Also .

Besides,

so that , and also . But for any and , we have

and the analogues for v ∙ u. We conclude that

Now, for linearly independent elements , let with n(u₁, v) ≠ 0 = n(u₂, v). Then the alternative law gives (u₁ ∙ u₂)∙ v = −(u₁ ∙ v)∙ u₂ +u₁ ∙ (u₂ ∙ v + v ∙ u₂) = −n(u₁, v)u₂ ≠ 0, so that u₁ ∙ u₂ ≠ 0. In particular, , and the same happens with .

Consider the trilinear map:

This is alternating because x^∙2 = 0 for any by the Cayley–Hamilton equation, and n(x ∙ y, y) = −n(x, y ∙ y) = 0. It is also non-zero, because , so that .

Fix a basis {u₁, u₂, u₃} of with n(u₁ ∙ u₂, u₃₎ = 1 and take v₁ := u₂ ∙ u₃, v₂ := u₃ ∙ u₁, v₃ := u₁ ∙ u₂. Then {v₁, v₂, v_3} is the dual basis in relative to the norm, and the multiplication of the basis {e₁, e₂, u₁, u₂, u₃, v₁, v₂, v₃} is completely determined. For instance, v₁ ∙ v₂ = v₁ ∙ (u₃ ∙ u₁₎ = −u₃ ∙ (v₁ ∙ u₁₎ = −u₃ ∙ (−n(v₁, u₁₎e₂) = u₃, …

The multiplication table is given in Figure 2.1.

The Cayley algebra with this multiplication table is called the split Cayley algebra and denoted by . The subalgebra spanned by e₁, e₂, u₁, v₁ is isomorphic to the algebra of 2 × 2 matrices.

We summarize the above arguments in the next result.

THEOREM 2.2.– There are, up to isomorphism, only three Hurwitz algebras with isotropic norm: × , and .

COROLLARY 2.3.– The real Hurwitz algebras are, up to isomorphism, the following algebras:

 – the classical division algebras ℝ, ℂ, ℍ, and ;

 – the algebras ℝ × ℝ, and .

Figure 2.1. Multiplication table of the split Cayley algebra

PROOF.– It is enough to take into account that a non-degenerate quadratic form over ℝ is either isotropic or definite. Hence, the norm of a real Hurwitz algebra is either isotropic or positive definite (as n(1) = 1). □