We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra.

We can also define the algebra of octonions $\mathbb O$ over any field $k$, and if over $k$ the $8$-ary quadratic form $Q=x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2$ is anisotropic, then $\mathbb O$ is again a division algebra —a non-associative one, but oh well.

What happens to $\mathbb O$ if $k$ is such that $Q$ is isotropic?

The classical structure theory of non-commutative Jordan algebras tells us that $\mathbb O$ is, over any field, a direct product of simple flexible power-associative algebras coming from a rather restricted list: simple commutative Jordan algebras, quasiassociative algebras, and flexible quadratic algebras with nondegenerate norm forms (Shafer's book on non-associative algebra explains all this, which is —I'd say— mostly forgotten nowadays) but I am pretty sure one can be very specific about what comes out in the case of octonions. In other words, one can probably find something playing the role of «matrix algebra» in the statement about quaternions.

**N.B.** All this is over fields of characteristic zero.

isa difference of complexity in going from quaternions to matrix algebras, which is not found in going to the split-octonions (at least, in the references to these that I found) $\endgroup$ – Mariano Suárez-Álvarez Mar 31 '13 at 4:29