In the mathematical field of Lie theory, the radical of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the largest solvable ideal of ${\displaystyle {\mathfrak {g}}.}$^[1]

The radical, denoted by ${\displaystyle {\rm {rad}}({\mathfrak {g}})}$, fits into the exact sequence

${\displaystyle 0\to {\rm {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})\to 0}$.

where ${\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}$ is semisimple. When the ground field has characteristic zero and ${\displaystyle {\mathfrak {g}}}$ has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of ${\displaystyle {\mathfrak {g}}}$ that is isomorphic to the semisimple quotient ${\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}$ via the restriction of the quotient map ${\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}}).}$

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