Start with the

field of

real numbers

**R**. The first generalisation
of these was the field of

complex numbers which is 2-dimensional over

**R**.
Next came Hamilton's

quaternions **H** which
are 4-dimensional over

**R**. The quaternions are no longer a field
but they are still a

division ring.

In 1845 Cayley
made a further generalisation when he created the
octonions **O**. (In fact they were also independently invented
by J.J. Graves slightly earlier.)
This time **O** is 8-dimensional over **R**.
A typical octonion is a 2x2 matrix

-- --
| a r |
| s b |
-- --

where

*a,b* are real numbers and

*r,s* are

vectors in

**R**^{3}. Obviously the usual 2x2 matrix

multiplication
won't work here. But we can define a multiplication by the following rule:

-- -- -- -- -- --
| a r || c t | = | ac - r.v at + dr +sxv |
| s b || v d | | sc + bv + rxt bd - s.t |
-- -- -- -- -- --

Here, we are making use of two operations on

**R**^{3},
the

dot product,

*(p,q) -> p.q* and the

cross product,

*(p,q) -> pxq*.

In **O** most of the axioms for a ring still hold (the identity
element is the 2x2 identity matrix) and each nonzero element
has an inverse but associativity does not hold. The main interest in the
octonions is that they can be used
to construct **g**_{2},
an important non-classical Lie algebra. (Note that the same construction
will work with any base field replacing **R**.)