 Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra formula_1 is the largest solvable ideal of formula_2

The radical, denoted by formula_3, fits into the exact sequence
where formula_5 is semisimple.When the ground field has characteristic zero and formula_6 has finite dimension, then Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of formula_6 that is isomorphic to the semisimple quotient formula_8 via the quotient map formula_9

A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.