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The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of (real) dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group of order 2 generated by complex conjugation.
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As well as the complex Lie group of type E8, there are three real forms of the group, all of real dimension 248, as follows:
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* A compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
* A split form, which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
* A third form, which has maximal compact subgroup E7×SU(2)/(-1×-1), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
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For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.
