# Octonion

Le octoniones es un normate algebra de division a 8 dimensiones super le corpore ${\displaystyle \mathbb {R} }$ del numeros real e un extension del quaterniones. Pro le insimul del octoniones, le symbolo ${\displaystyle \mathbb {O} }$ es usate. Le prime vice, illos es describite in 1843 per John Thomas Graves in un littera a William Rowan Hamilton, ma publicate duo annos plus tarde in 1845 per Arthur Cayley.

## Un possibile tabella de multiplication

Per medio del notation del octoniones unitate in le forma ${\displaystyle \{e_{0},e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\}}$  con le elemento scalar ${\displaystyle e_{0}=1\in \mathbb {R} }$ , su tabella de multiplication es assi:

${\displaystyle e_{i}e_{j}}$  ${\displaystyle e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$ ${\displaystyle e_{j}}$ ${\displaystyle e_{i}}$ ${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{3}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{5}}$ ${\displaystyle -e_{4}}$ ${\displaystyle -e_{7}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{2}}$ ${\displaystyle -e_{3}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{4}}$ ${\displaystyle -e_{5}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ ${\displaystyle -e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{6}}$ ${\displaystyle e_{5}}$ ${\displaystyle -e_{4}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{5}}$ ${\displaystyle -e_{6}}$ ${\displaystyle -e_{7}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{7}}$ ${\displaystyle e_{6}}$ ${\displaystyle -e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle -e_{3}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{5}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle -e_{0}}$ ${\displaystyle -e_{1}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{6}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{3}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{1}}$ ${\displaystyle -e_{0}}$