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It is well-known the Hurwitz theorem showing that every unital composition algebra is of dimension 1, 2, 4 or 8 over the base field.

Is there an analogue in the non unital case? ┬┐Do non unital composition algebras of arbitrary dimension exist?

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I think a big question is how you want to axiomatize these structures, as two axiomatizations which are equivalent in the presence of a unit might become drastically different when you remove from both the assumption of a unit. Particularly, the question is whether you want a notion of conjugation still to exist (in standard axiomatizations, as for example in the book by John H. Conway and Derek Smith, conjugation is defined by $v \mapsto 2\langle v, e \rangle - v$). –  Todd Trimble Sep 10 '12 at 14:38
    
Argh, $v \mapsto 2\langle v, e \rangle e - v$, where $e$ is the unit. –  Todd Trimble Sep 10 '12 at 14:39
    
No, I would not need a conjugation. My definition for composition algebra is (cf.Zhevlakov et al. for instance): An algebra over a field F with quadratic form n(x) is called a composition algebra if: -n(xy)=n(x)n(y). -The form n(x) is strictly nondegenerate. - A has identity. In particular I am interested in removing the 2 las conditions so I just have a quadratic form n(x) [f(x,y)=n(x+y)-n(x)-n(y) bilinear] admitting composition. –  Antonio Oller Sep 10 '12 at 14:56
    
Sorry not to have replied sooner. If the quadratic form can be degenerate, what stops us from having both the algebra multiplication and $n$ be identically zero? –  Todd Trimble Sep 15 '12 at 17:05
    
And what if we keep non-degeneracy but remove the unit? –  Antonio Oller Sep 27 '12 at 18:58

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