Radical of a Lie algebra
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.
- A Lie algebra is semisimple if and only if its radical is .
- A Lie algebra is reductive if and only if its radical equals its center.
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