Let $G$ be a simply connected Lie group. If $G$ is nilpotent, we know every connected Lie subgroup of $G$ is simply connected by Baker-Campbell-Hausdorff formula. What about the converse? If every connected Lie subgroup $G$ is simply connected, then is $G$ nilpotent? If not, can we classify them?
2026-02-23 02:24:22.1771813462
Connected Lie group for which every connected Lie subgroup is simply connected
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Consider the group of matrices $\pmatrix{a &b\cr 0&c}, a,c>0$ every connected subgroup of this group is simply connected but this group is solvable and not nilpotent.