Zorn vector-matrix description of octonion multiplication Zorn's vector-matrices are a way to describe split octonions by treating them as matrices
$$ \begin{bmatrix}a & \mathbf v\\ \mathbf w & b\end{bmatrix} $$
where $a,b \in \mathbb{R}$ and $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$.  The multiplication of split octonions is then given by
$$ \begin{bmatrix}a & \mathbf v\\ \mathbf w & b\end{bmatrix} \begin{bmatrix}a' & \mathbf v'\\ \mathbf w' & b'\end{bmatrix} = \begin{bmatrix}aa' + \mathbf v\cdot\mathbf w' & a\mathbf v' + b'\mathbf v + \mathbf w \times \mathbf w'\\ a'\mathbf w + b\mathbf w' - \mathbf v\times\mathbf v'  & bb' + \mathbf v'\cdot\mathbf w \end{bmatrix} $$
I think there's a similar formula that describes the multiplication of ordinary octonions: to get it, we probably just change some signs above.   Does anyone know this formula, or a reference to it?
I should probably work it out myself, but I feel someone must have already have written it down.
 A: I know of two vector-matrix-like representations of the octonions, but unfortunately none of them rely on changing the multiplication.
One of them is to realize the octonions as a subalgebra of the complex octonions $\mathbb{C} \otimes \mathbb{O}$ (this is a split algebra since $\mathbb{C}$ is algebraically closed, and thus can be identified with Zorn's algebra with complex entries). The realization of $\mathbb{O} \subset \mathbb{C} \otimes \mathbb{O}$ is given by vector-matrices of the form
$$\begin{pmatrix} a & \mathbf{v} \\ -\bar{\mathbf{v}} & \bar{a} \end{pmatrix},$$
where $a$ and the components of $\mathbf{v}$ are now complex numbers, and the bar denotes conjugation. The multiplication is formally the same. This is analogous to the known representation of quaternions as ordinary $2\times 2$ complex matrices of the form
$$\begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix}.$$
Intuitively, this is similar to the idea of finding the compact Lie groups inside the complex Lie groups by imposing some "unitarity condition" (in this case the condition is that $M M^{\dagger}$ be a real multiple of the identity, where $\dagger$ denotes the usual Hermitian conjugation, extended in the obvious way to vector-matrices).

The other way is to realize $\mathbb{O}$ inside the split-sedenions (the Cayley-Dickson doubling of the split-octonions). These are isomorphic to vector-matrices where the vectors are $7$-dimensional:
$$\begin{pmatrix} a & \mathbf{V} \\ \mathbf{W} & b \end{pmatrix},$$
where I use uppercase to denote seven-dimensional vectors. The multiplication is again given by the same formula, except that now $\times$ is the seven-dimensional cross product. (I don't know of a reference for this, but it's easy to find an explicit isomorphism). The subalgebra given by
$$\begin{pmatrix} a & \mathbf{V} \\ -\mathbf{V} & a \end{pmatrix}$$
is then isomorphic to $\mathbb{O}$, as can be explicitly checked. There are analogous representations of quaternions and complex numbers given by
$$\begin{pmatrix} a & \mathbf{v} \\ -\mathbf{v} & a \end{pmatrix}, \quad \begin{pmatrix} a & b \\ -b & a \end{pmatrix}$$
respectively (the latter is well-known).
This representation doesn't require the use of complex entries, though on the other hand it feels a bit circular to me, since the 7D cross product is already closely related to octonions; if you "know" one you already "know" the other in some sense.

I don't know if there is a natural way to arrive at the octonions by modifying only the multiplication formula instead of the vector-matrices themselves, but it strikes me as somewhat unlikely. This would be akin to realizing the quaternions as $2\times 2$ real matrices with a modified multiplication. If there is some way of doing it, it would probably be more involved than a change of signs.
