Editing G2 (mathematics) (section)

== History ==

The Lie algebra <math>\mathfrak{g}_2</math>, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras.  On May 23, 1887, [[Wilhelm Killing]] wrote a letter to [[Friedrich Engel (mathematician)|Friedrich Engel]] saying that he had found a 14-dimensional simple Lie algebra, which we now call <math>\mathfrak{g}_2</math>.<ref>{{cite journal
 | last = Agricola | first = Ilka | author-link = Ilka Agricola
 | issue = 8
 | journal = Notices of the American Mathematical Society
 | mr = 2441524
 | pages = 922–929
 | title = Old and new on the exceptional group ''G''<sub>2</sub>
 | url = https://www.ams.org/notices/200808/tx080800922p.pdf
 | volume = 55
 | year = 2008}}</ref>

In 1893, [[Élie Cartan]] published a note describing an open set in <math>\mathbb{C}^5</math> equipped with a 2-dimensional [[distribution (differential geometry)|distribution]]—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra <math>\mathfrak{g}_2</math> appears as the infinitesimal symmetries.<ref>{{cite journal|author=Élie Cartan|title=Sur la structure des groupes simples finis et continus|journal=C. R. Acad. Sci.|volume=116|year=1893|pages=784–786}}</ref> In the same year, in the same journal, Engel noticed the same thing.   Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball.   The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.<ref>{{cite journal| title = G<sub>2</sub> and the "rolling distribution" | author = Gil Bor and Richard Montgomery |journal =L'Enseignement Mathématique|volume =55|year=2009|pages=157–196|doi=10.4171/lem/55-1-8|arxiv=math/0612469| s2cid = 119679882 }}</ref><ref>{{cite journal| title = G<sub>2</sub> and the rolling ball | author = John Baez and John Huerta |arxiv=1205.2447|journal =Trans. Amer. Math. Soc.|volume =366| issue = 10 |year=2014|pages=5257–5293|doi=10.1090/s0002-9947-2014-05977-1}}</ref>

In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G<sub>2</sub>.<ref>{{cite journal|author=Friedrich Engel|title=Ein neues, dem linearen Komplexe analoges Gebilde|journal=Leipz. Ber.|volume=52|year=1900|pages=63–76,220–239}}</ref>

In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group.<ref>{{cite book|author=Élie Cartan|chapter= Nombres complexes|title=Encyclopedie des Sciences Mathematiques|publisher=Gauthier-Villars|location=Paris|year= 1908|pages = 329–468}}</ref>  In 1914 he stated that this is the compact real form of G<sub>2</sub>.<ref>{{citation|author=Élie Cartan|title=Les groupes reels simples finis et continus|journal=Ann. Sci. École Norm. Sup.|volume=31|year=1914|pages=255–262}}</ref>

In older books and papers, G<sub>2</sub> is sometimes denoted by E<sub>2</sub>.