The most elementary way to construct the octonions is to give their
multiplication table. The octonions are an 8-dimensional algebra
with basis
,
and their multiplication is given in this table, which describes
the result of multiplying the element in the th row by the
element in the th column:

Table 1 — Octonion Multiplication Table

Unfortunately, this table is almost completely unenlightening! About the only
interesting things one can easily learn from it are:

are square roots of -1,

and anticommute when :

the index cycling identity holds:

where we think of the indices as living in , and

the index doubling identity holds:

Together with a single nontrivial product like , these
facts are enough to recover the whole multiplication table. However, we
really want a better way to remember the octonion product. We should
become as comfortable with multiplying octonions as we are with
multiplying matrices! And ultimately, we want a more conceptual
approach to the octonions, which explains their special properties and
how they fit in with other mathematical ideas. In what follows, we give
some more descriptions of octonion multiplication, starting with a nice
mnemonic, and working up to some deeper, more conceptual ones.