Editing Outer automorphism group (section)

== In complex and real simple Lie algebras ==
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, {{mvar|𝔤}}, the automorphism group {{math|Aut(''𝔤'')}} is a [[semidirect product]] of {{math|Inn(''𝔤'')}} and {{math|Out(''𝔤'')}}; i.e., the [[exact sequence|short exact sequence]]
: {{math|1 ⟶ Inn(''𝔤'') ⟶ Aut(''𝔤'') ⟶ Out(''𝔤'') ⟶ 1}}
splits. In the complex simple case, this is a classical result,<ref>{{citation| last1         = Fulton
| first1        = William
| author1-link  = William Fulton (mathematician)
| last2        = Harris
| first2       = Joe
| author2-link = Joe Harris (mathematician)
| year         = 1991
| title        = Representation theory. A first course
| publisher    = Springer-Verlag
| location     = New York
| series       = [[Graduate Texts in Mathematics]], Readings in Mathematics
| volume       = 129
| isbn         = 978-0-387-97495-8
| doi          = 10.1007/978-1-4612-0979-9
| oclc         = 246650103
| language     = en-gb
| mr           = 1153249|contribution=Proposition D.40}}</ref> whereas for real simple Lie algebras, this fact was proven as recently as 2010.<ref name="JOLT">[http://www.heldermann.de/JLT/JLT20/JLT204/jlt20035.htm JLT20035]</ref>