Editing Root system (section)

==History==
The concept of a root system was originally introduced by [[Wilhelm Killing]] around 1889 (in German, ''Wurzelsystem''<ref>{{harvnb|Killing|1889}}</ref>).<ref name=Bourbaki98p270>{{harvnb|Bourbaki|1998|p=270}}</ref> He used them in his attempt to classify all [[simple Lie algebra]]s over the [[field (mathematics)|field]] of [[complex number]]s. (Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F<sub>4</sub>. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.<ref>{{harvnb|Coleman|1989|p=34}}</ref>)

Killing investigated the structure of a Lie algebra <math>L</math> by considering what is now called a [[Cartan subalgebra]] <math>\mathfrak{h}</math>. Then he studied the roots of the [[characteristic polynomial]] <math>\det (\operatorname{ad}_L x - t)</math>, where <math>x \in \mathfrak{h}</math>. Here a ''root'' is considered as a function of <math>\mathfrak{h}</math>, or indeed as an element of the dual vector space <math>\mathfrak{h}^*</math>. This set of roots forms a root system inside <math>\mathfrak{h}^*</math>, as defined above, where the inner product is the [[Killing form]].<ref name=Bourbaki98p270/>