2. Constructing the Octonions

The most elementary way to construct the octonions is to give their multiplication table. The octonions are an 8-dimensional algebra with basis $1, e_1,e_2,e_3,e_4,e_5,e_6,e_7$, and their multiplication is given in this table, which describes the result of multiplying the element in the $i$th row by the element in the $j$th column:

	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_6$	$e_7$
$e_1$	$-1$	$e_4$	$e_7$	$-e_2$	$e_6$	$-e_5$	$-e_3$
$e_2$	$-e_4$	$-1$	$e_5$	$e_1$	$-e_3$	$e_7$	$-e_6$
$e_3$	$-e_7$	$-e_5$	$-1$	$e_6$	$e_2$	$-e_4$	$e_1$
$e_4$	$e_2$	$-e_1$	$-e_6$	$-1$	$e_7$	$e_3$	$-e_5$
$e_5$	$-e_6$	$e_3$	$-e_2$	$-e_7$	$-1$	$e_1$	$e_4$
$e_6$	$e_5$	$-e_7$	$e_4$	$-e_3$	$-e_1$	$-1$	$e_2$
$e_7$	$e_3$	$e_6$	$-e_1$	$e_5$	$-e_4$	$-e_2$	$-1$

Table 1 — Octonion Multiplication Table

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

• $e_1,\dots,e_7$ are square roots of -1,
• $e_i$ and $e_j$ anticommute when $i \ne j$:
  
  $$e_i e_j = -e_j e_i$$
  
  

• the index cycling identity holds:
  
  $$% latex2html id marker 1527 e_i e_j = e_k \; \implies \; e_{i+1} e_{j+1} = e_{k+1}$$
  
  
  where we think of the indices as living in $\Z_7$, and
• the index doubling identity holds:
  
  $$% latex2html id marker 1528 e_i e_j = e_k \; \implies \; e_{2i} e_{2j} = e_{2k} .$$
  
  

Together with a single nontrivial product like $e_1 e_2 = e_4$, 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.

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