Frobenius from Hurwitz's theorem Can we deduce Frobenius theorem from Hurwitz's theorem on normed division algebras?
The Frobenius theorem states that the only associative finite dimensional division algebras over the real numbers are $\mathbb R$, $\mathbb C$, and $\mathbb H$ (the quaternions).
Hurwitz's theorem states that the only normed division algebras over the real numbers are $\mathbb R$, $\mathbb C$, $\mathbb H$ (the quaternions), and algebra $\mathbb O$ of octonions.
thanks.
 A: Division algebras are amazing as adding some other assumption reduces the class of
objects to a small set:


*

* Frobenius theorem (1879): associative real finite dimensional division algebras:   R,C,H

* Wedderburn theorem (1905): All finite division algebras are finite fields

* Hurwitz theorem (1922): the normed real division algebras are    R,C,H,O

* Mazur theorem (1938): the real Banach division algebras are  R,C,H.

* Gelfand theorem (1939): the commutative Banach division algebras are  R,C



*

* Gelfand-Mazur theorem: the only complex Banach division algebra is  C

*Frobenius-Hurwitz theorem: complete associative division algebras are C,H


Each theorem is different:


*

*Frobenius assumes "finite dimensional and associative"

*Wedderburn assumes "finite"

*Hurwitz assumes "normed"

*Mazur assumes "Banach"

*Gelfand assumes "commutative"



*

* Gelfand-Mazur assumes "complex and Banach" (nomenclature by Rudin Fun.Anal. 1973)

* Frobenius-Hurwitz is a merger of nomenclature and a consequence of the two theorems, for non-abelian
algebras, completeness has to be defined properly and as done by Eilenberg and Niven
in 1944. So, without that definition, the term complete would not make sense.


It is natural to ask for a relation between the Frobenius and Hurwitz theorem.
There seems no obvious link between the Frobenius and Hurwitz statements, as the class of
"normed division algebras" and "division algebras" is different and
associativity is a strong assumption. The question asks whether there
could be dependencies between the proofs of the theorems. The answer is probably "no"
as it appears not easy: going from Frobenius to Hurwitz requires getting rid of
the attributes "associative, finite dimensional" and adding "normed" instead.
If one assumes the Frobenius theorem and adds the attribute "normed", then one still
has R,C,H. Why it is true that getting rid of "associative" only will add 
the Octonion algebra? A short cut of Hurwitz via Frobenius of course would still be nice. 
Historically, Hurwitz theorem was only published posthumously. Hurwitz paper does not
cite Frobenius. Hurwitz (1859-1919)  and Frobenius (1849-1917) were contemporaries
and both worked in Zurich. Frobenius was at ETH Zuerich between 1875 and 1892. When
Frobenius took over Kroecker's chair in Berlin, Hurwitz in turn took over his chair at ETH
from 1892 until his death 1919. There is not only a mathematical affinity
through these two theorems but also a personal story between these two mathematicians.
Their names even cross in the form of Hurwitz-Frobenius manifolds.
