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If the nilradical () of the Lie algebra  of a pro-Lie group  G  is finite dimensional modulo the center (), then every identity neighborhood  U  of  G  contains a closed normal subgroup  N  such that  G/N  is a Lie group and  G  and  N  ×  G/N  are locally isomorphic.

Iwasawa's Local Splitting Theorem for Pro-Lie Groups