Solvable radical

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. Its solvable radical, denoted $\mathrm{rad}(\mathfrak{g})$, is the largest solvable ideal of $\mathfrak{g}$: it is a solvable ideal and contains every solvable ideal of $\mathfrak{g}$. Here solvability is as in Derived series and solvable Lie algebras, and “ideal” has the meaning in Lie subalgebras, ideals, and center.

Existence and uniqueness are not assumed merely from the phrase “largest.” They are supplied by The sum of solvable ideals is solvable ↗, which proves that finite sums remain solvable and that finite dimensionality reduces the sum of all solvable ideals to a finite sum.

Thus $\mathrm{rad}(0)=0$. If $\mathfrak{g}$ is solvable, then $\mathrm{rad}(\mathfrak{g})=\mathfrak{g}$; otherwise the radical may be zero or a proper nonzero ideal. No characteristic assumption is made.