High Quality Content by WIKIPEDIA articles! In mathematics, E8 is the name given to several closely related exceptional simple Lie groups and Lie algebras of dimension 248; the same notation is sometimes used for their root lattice, which has rank 8. Wilhelm Killing discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by Elie Cartan. Cartan also classified the three real forms of the Lie algebra, which correspond to 5 different real Lie groups of dimension 248, one of which is compact. There are also forms of the E8 groups and Lie algebras over other fields; for example, the E8 groups over finite fields form one of the infinite series of finite simple groups. The designation E8 comes from Killing and Cartan's classification of the complex simple Lie algebras, which fall into four infinite families labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.

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