A nilpotent Lie group is a Lie group G which is connected and whose Lie algebra is a nilpotent Lie algebra g. That is, its Lie algebra lower central series g_1 = [g, g], g_2 = [g, g_1], ... eventually vanishes, g_k = 0 for some k. So a nilpotent Lie group is a special case of a solvable Lie group.
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