Let be a field and let be a finite-dimensional Lie algebra over . There exists a unique maximal solvable ideal, called the radical, for the following reason.
Firstly let and be two solvable ideals of . Then is again an ideal of , and it is solvable because it is an extension of by . Now consider the sum of all the solvable ideals of . It is nonempty since is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal.