radical

Let $𝔤$ be a Lie algebra. Since the sum of any two solvable ideals of $𝔤$ is in turn solvable, there is a unique maximal solvable ideal of any Lie algebra. This ideal is called the radical of $𝔤$. Note that $𝔤/\mathrm{rad}𝔤$ has no solvable ideals, and is thus semi-simple. Thus, every Lie algebra is an extension (http://planetmath.org/Extension) of a semi-simple algebra by a solvable one.