# Classification of subalgebras of composition algebras

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?)

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.