Editing Octonion (section)

===Automorphisms===
An [[automorphism]], {{mvar|A}}, of the octonions is an invertible [[linear transformation]] of <math>\ \mathbb{O}\ </math> which satisfies

:<math>A(xy) = A(x)\ A(y) ~.</math>

The set of all automorphisms of <math>\ \mathbb{O}\ </math> forms a group called {{nobr|{{math|[[G2 (mathematics)|''G''{{sub|2}}]]}} .}}<ref>{{harv|Conway|Smith|2003|loc=ch&nbsp;8.6}}</ref> The group {{math|''G''{{sub|2}} }} is a [[simply connected]], [[Compact group|compact]], real [[Lie group]] of dimension&nbsp;14. This group is the smallest of the exceptional Lie groups and is isomorphic to the [[subgroup]] of {{math|Spin(7)}} that preserves any chosen particular vector in its 8&nbsp;dimensional real spinor representation. The group {{math|Spin(7)}} is in turn a subgroup of the group of isotopies described below.

''See also'': {{math|[[PSL(2,7)]]}} – the [[automorphism group]] of the Fano plane.