In the context of Lie theory, a **Borel subalgebra** is a type of subalgebra of a Lie algebra that has certain important properties. Specifically, for a complex semisimple Lie algebra \(\mathfrak{g}\), a Borel subalgebra is a maximal solvable subalgebra.
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For a choice of positive roots, the corresponding Borel subalgebra is , the sum of a Cartan subalgebra and all positive root spaces.