Matrix Representation of Octonions Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following

$$   1 \mapsto \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{pmatrix},\quad \mathrm i \mapsto \begin{pmatrix} \mathrm i_{\mathbb C} &amp; 0 \\ 0 &amp; -\mathrm i_{\mathbb C} \end{pmatrix},\quad \mathrm j \mapsto \begin{pmatrix} 0 &amp; 1 \\ -1 &amp; 0 \end{pmatrix},\quad \mathrm k \mapsto \begin{pmatrix} 0 &amp; \mathrm i_{\mathbb C} \\ \mathrm i_{\mathbb C} &amp; 0 \end{pmatrix}, 
$$
I always wondered if there is also matrix representation of the octonions?

How is the non-associativity realised with matrices?
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.
I tried to convert different dimensional base units to matrices. The reals are trivial, the complexes are easy, the quaternions are interesting, but solvable and the octonions are so tricky, that it does not work. My matrix looks like this:
$ M_{n m} = i_x^{o-1} o (i_n i_m)$
So e. g.
M = $ \begin{pmatrix} 0 &amp; -1 &amp; -2 &amp; -3 \\ 1 &amp; 0 &amp; -3 &amp; 2 \\ 2 &amp; 3 &amp; 0 &amp; -1 \\ 3 &amp; -2 &amp; 1 &amp; 0\end{pmatrix}$
for the quaternions, it can be for the octonions, too.
But to make sense of all these matrices, we should convert it to another (in this case 4) matrices with full of 0, 1 and -1 numbers. And if we multiplicate them, we get each other, right? I tested it, and it works at quaternions, but so does not at octonions. I am really sorry.
Maybe the multiplication table of octonions is wrong or the previous person was right talking about its problematics of the associativity.