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Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ matrices (or split-quaternions) $M_2(F)$, and Zorn's vector-matrix algebra (or split-octonions) $Zo(F)$. The elements of these four algebras admit a respective description in terms of $2\times 2$ matrices or generalized versions of such, namely:

$$\begin{pmatrix} a & 0 \\ 0 & a\end{pmatrix}, \begin{pmatrix} a & 0 \\ 0 & b\end{pmatrix}, \begin{pmatrix} a & b \\ c & d\end{pmatrix}, \begin{pmatrix} a & (b,c,d) \\ (e,f,g) & h\end{pmatrix},$$

where $a,b,\ldots,h$ are arbitrary elements of $F$, and in the last case we use the modified matrix multiplication described in this article, which is nonassociative.

All four algebras are clearly subalgebras of the biggest one $Zo(F)$, if we identify any $x$ in the upper-right and lower-left corners with the vector $(x,0,0)$. There exist other subalgebras (necessarily not composition), namely the dual numbers $F[\varepsilon]$ where $\varepsilon^2=0$, the upper triangular $2\times 2$ matrices $B_2(F)$ (apparently also called ternions), and the sextonions $Se(F)$. A matrix representation for these subalgebras is respectively

$$\begin{pmatrix} a & b \\ 0 & a\end{pmatrix}, \begin{pmatrix} a & b \\ 0 & c\end{pmatrix}, \begin{pmatrix} a & (b,c,0) \\ (d,0,e) & f\end{pmatrix}$$

(for the last one see here). There is yet another subalgebra that I found by trial and error, that I guess we could call the "quintonions" $Qui(F)$. A matrix representation is

$$\begin{pmatrix} a & (b,0,0) \\ (0,c,d) & e\end{pmatrix}.$$

My question is

Is this list exhaustive? Do there exist any other subalgebras of composition algebras (unital, properly containing $F$ as a subalgebra) not isomorphic to the ones already described ($F$, $F[\varepsilon]$, $F\oplus F$, $B_2(F)$, $M_2(F)$, $Qui(F)$, $Se(F)$, $Zo(F)$)?

(By the way, have these quintonions been described anywhere in the literature?)

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I finally found an answer online, I'll post it here in case someone finds it of value. The classification above is incomplete: there exist three more subalgebras of $Zo(F)$. I had missed them because I foolishly thought that all subalgebras of the same dimension were isomorphic.

Subalgebras of the split-octonions over $\mathbb{R}$ were classified two years ago in this paper. We can get the corresponding classification over its algebraic closure $\mathbb{C}$ by tensoring with it (thus identifying some of the algebras).

Discarding the completely nilpotent ones (since they don't properly contain the base field as required), and if I didn't make any mistake, we recover all algebras described in the question, plus three more. The new ones are (note that although the paper originally deals with $\mathbb{R}$, the following constructions are valid over any field):

  • The three-dimensional "bidual numbers" $F[\varepsilon, \varepsilon']$ with two orthogonal idempotents $\varepsilon^2 = \varepsilon'^2 = \varepsilon\varepsilon'=0$.

  • The four-dimensional Grassmann algebra $\Lambda(F^2)$.

  • The "biduals" over $F\oplus F$, i.e. $(F\oplus F)[\varepsilon, \varepsilon']$, another four-dimensional algebra.

A matrix representation is given by

$$\begin{pmatrix} a & (b,0,0) \\ (0,c,0) & a\end{pmatrix}, \begin{pmatrix} a & (b,0,0) \\ (0,c,d) & a\end{pmatrix}, \begin{pmatrix} a & (b,0,0) \\ (0,c,0) & d\end{pmatrix}$$

respectively. Interestingly, none of them can be embedded in $M_2(F)$ despite being associative.

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