Matrix Representation of Octonions

A $\mathbb R$-linear function $\phi:\mathbb O\to A$ to any real associative algebra which is multiplicative and unitary (so that $\phi(xy)=\phi(x)\phi(y)$ for all $x$, $y\in \mathbb O$, and $\phi(1)=1$) has to vanish on the bilateral ideal $I$ generated by the elements of the form $$(x\cdot y)\cdot z-x\cdot(y\cdot z)$$ with $x$, $y$, $z\in\mathbb O$. Now, this ideal is not zero because $\mathbb O$ is not associative, and therefore $I=\mathbb O$, because $\mathbb O$ has no non-trivial bilateral ideals (it is a division thing).

It follows that the map $\phi$ is in fact zero.


M. Zorn (Abh.Mat.Sem.Hamburg,9,395, 1933) modified matrix multiplication for 2x2 "vector-matrices" with two real-numbers as diagonal entries and two 3d-vectors off the diagonal.

See also Matrix Representation of Octonions and Generalizations by Jamil Daboul and Robert Delbourgo, (9 June 1999)


We have two Octonions algebra, one is a division algebra (with no zero divisors) and a split algebra (with zero divisors). Both are constructed by Cayley-dicson process over a field $F$. If you pick the split algebra of octonions you can show that it is isomorphic to a non-assossiative matrix algebra called Zorn matrix-vector algebra, $M(F)$ where the elements are: $\left[ \begin{array}{cc} a &\textbf{x}\\ \textbf{y} & b \end{array}\right]$, where $\textbf{x,y}\in F^3$.

The addition is defined term by term and the multiplication defined as $\left[ \begin{array}{cc} a_1 &\textbf{x}_1\\ \textbf{y}_1 & b_1 \end{array}\right] \left[ \begin{array}{cc} a_2 &\textbf{x}_2\\ \textbf{y}_2 & b_2 \end{array}\right] = \left[ \begin{array}{cc} a_1a_2+\textbf{x}_1.\textbf{y}_2 & a_1\textbf{x}_2+b_2\textbf{x}_1-\textbf{y}_1\wedge\textbf{y}_2\\ a_2\textbf{y}_1+b_1\textbf{y}_2+\textbf{x}_1\wedge\textbf{x}_2 & b_1b_2+\textbf{y}_1.\textbf{x}_2 \end{array}\right]$. Where $\wedge$ denote the exterior product over $F^3$.

It is easy to show that it is non-assossiative. And, as we have the general linear group, the special linear group and the projective linear group over the assossiative matrix algebra, here we have the concept of general linear loop, special linear loop and projective linear loop over M(K).