Talk:Frobenius theorem (real division algebras)

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 Field: Algebra

Is it possible someone would be able to include a proof for this?

Serious mistake in proof[edit]

If e_1, \ldots , e_n is orthonormal basis, then e_ie_j = 0 by definition of orthonormality, so the claim e_ie_j = -e_je_i is wrong (actually not completely wrong, but doesn't make any sense since e_ie_j = 0) and the following arguments about quaternions and case n>2 are also wrong.

Also there's a type in case n=2: e_1e_2 = - e_1e_2, it should be e_1e_2 = - e_2e_1 for the case of quaternions, but it would contradict with orthogonality. — Preceding unsigned comment added by (talk) 11:03, 3 October 2012 (UTC)

There is no mistake: Orthonormality says that the inner product B(e_1,e_2) is zero, not that the algebra product e_1e_2 is zero. The definition of the inner product is  B(e_1.e_2):=  -e_1e_2-e_2e_1. Thus e_1e_2=-e_2e_1. Mike Stone (talk) 15:32, 11 June 2013 (UTC)