Abstract
The surjectivity of the exponential function of complex algebraic, in particular of complex semisimple Lie groups, and of complex splittable Lie groups is equivalent to the connectedness of the centralizers of the nilpotent elements in the Lie algebra. This implies that the only complex semisimple Lie groups with surjective exponential function are isomorphic to finite products of the adjoint groups of SL(n,2).
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