History
The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem). He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.
Killing investigated the structure of a Lie algebra, by considering (what is now called) a Cartan subalgebra . Then he studied the roots of the characteristic polynomial, where . Here a root is considered as a function of, or indeed as an element of the dual vector space . This set of roots form a root system inside, as defined above, where the inner product is the Killing form.
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