Abdenacer Makhlouf - Algebra and Applications 1

Finally, for any x , y , z , using again [2.5],

so that, using left-right symmetry:

[2.6]

With y = 1, this gives and hence the Cayley–Hamilton equation. However, and this shows x ∙ ( x ∙ y ) = x ∙2∙ y . Symmetrically we get ( y ∙ x ) ∙ x = y ∙ x ∙2. □

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= α 1 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 ).