Let's consider the split Cayley-Dickson algebra $C$ over an arbitrary field $F$ (It is well known that all split composition algebras having the same dimension over base field are isomorphic, e.g., all split Cayley-Dickosn algebras are isomorphic to Zorn's vector-matrix algebra).
The question is: what are the conditions for existence of associative division subalgebra $A$ of $C$ with $dim_F A = 4?$
Of course, it is necessary for $F$ not to be algebraically closed. It also seems to me that it's not possible for $A$ to be a field because of its dimension and $A$ being a composition algebra, so $F$ must be infinite by Wedderburn's theorem.
I'm also wondering if $A$ is unique, because there may be more than one division composition algebra of dimension $4$ over $F$, but I'm not sure that all of them can be proper subalgebras of $C$. Let's consider this subspace of $C$: $ V = \left\{\begin{pmatrix} 0 & 0 \\ \alpha & v \\ \end{pmatrix}|\ \alpha \in F, v \in F^3\right\}. $
It's easy to see that $dim_F V=4$ and $V \cap A =0$ because every element of $V$ is not invertible, so by dimension counting we have $V \oplus A = C$.
We also have $F \subset A$, so i think $A$ must look like $ A = \left\{\begin{pmatrix} \beta & u \\ \varphi(u) & \beta \\ \end{pmatrix}|\ \beta \in F, u \in F^3\right\}, $
where $\varphi$ is an invertible linear map of $F^3$ satisfying certain identities (for example, $\varphi(v) \cdot u = v \cdot \varphi(u) $ , where $\cdot$ is the ordinary dot product — it can be easily obtained by multiplying elements from $A$ and comparing results).
For example, quaternions are the only division subalgebra of split octonions, and they can be represented (with multiplication mentioned above) as $ \mathbb{H} = \left\{\begin{pmatrix} \alpha & v \\ -v & \alpha \\ \end{pmatrix}|\ \alpha \in \mathbb{R}, v \in \mathbb{R}^3\right\} $ , where $\varphi(v)=-v$.
So, that's another question: am I correct about the construction of these division subalgebras? If yes, what else can we say about $\varphi?$
Sorry for my poor English and LaTeX skills.
Thank you in advance.
Associative division subalgebras of split Cayley-Dickson algebra
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ring-theory
division-algebras