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Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

EDIT: This question now has an answer over there

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

EDIT: This question now has an answer over there

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other things of dimension 2, 4, and 8

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

EDIT: This question now has an answer over there

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

The finite-dimension division algebras over the reals are:

• $\Bbb R$: the reals (dimension 1)
• $\Bbb C$: the complex numbers (dimension 2)
• $\Bbb H$: the quaternions (dimension 4)
• $\Bbb O$: the octonions (dimension 8)
• some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?