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  "body": "\n## Summary\n\nA solvable Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[solvable group|0FXJ7YFMMX]]: its derived series reaches the trivial group after finitely many steps.\nEquivalently, its Lie algebra is a solvable Lie algebra.\nSolvable Lie groups are the counterpart of semisimple ones.\nBy the Levi decomposition, every connected Lie group is built from a solvable normal subgroup and a semisimple part.\nTypical examples are groups of invertible upper triangular matrices.\n\n## Explanations\n\n### L1 human\n\nA solvable Lie group is one that can be taken apart, in finitely many steps, into commutative layers.\nTake the commutators of the group, then the commutators of those, and so on: in a solvable group this process ends with the identity alone.\nThe group of maps $x \\mapsto a x + b$ of the real line with $a \u003e 0$ is an example.\nIts commutators are the translations $x \\mapsto x + b$, and these commute with each other, so the process stops after two steps.\n",
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