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In the mathematical field of Lie theory, the radical of a Lie algebra g {\displaystyle {\mathfrak {g}}} is the largest solvable ideal of g . {\displaystyle {\mathfrak {g}}.}
The radical, denoted by r a d {\displaystyle {\rm {rad}}} , fits into the exact sequence
where g / r a d {\displaystyle {\mathfrak {g}}/{\rm {rad}}} is semisimple. When the ground field has characteristic zero and g {\displaystyle {\mathfrak {g}}} has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a subalgebra of g {\displaystyle {\mathfrak {g}}} that is isomorphic to the semisimple quotient g / r a d {\displaystyle {\mathfrak {g}}/{\rm {rad}}} via the restriction of the quotient map g → g / r a d . {\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}.}
A similar notion is a Borel subalgebra, which is a maximal solvable subalgebra.