Levi decomposition theorem

Statement
The general form of the Levi decomposition theorem is that, under certain circumstances, a finite-dimensional Lie algebra or Lie group is expressible as an internal semidirect product of its radical (the unique largest solvable ideal or normal subgroup) and a semisimple subalgebra or subgroup.

Version for real Lie algebras
Suppose $$L$$ is a finite-dimensional Lie algebra over the field of real numbers $$\R$$. The Levi decomposition theorem states that $$L$$ has a decomposition as an internal semidirect product of finite-dimensional Lie algebras $$R,S$$, i.e.:

$$L = R \rtimes S$$

where $$R$$ is the radical of $$L$$ (i.e., the unique largest solvable ideal) and $$S$$ is a semisimple subalgebra of $$L$$. $$R$$ is uniquely determined as an ideal in $$L$$, but there may be more than one possibility for $$S$$. Viewed as a quotient, $$S$$ is termed the Levi factor of $$L$$. The different possible choices for $$S$$ as subalgebras of $$L$$ are termed Levi subalgebras.

Other versions
Below are some versions for groups and Lie algebras. In each version, the group in question is expressible as an internal semidirect product of the radical, which is the unique maximal (solvable normal subgroup or solvable ideal, depending on whether we are working with groups or Lie algebras) and another group/Lie algebra which is semisimple in a suitable sense. The semisimple quotient is termed the Levi factor, and particular choices of semisimple complements are termed Levi subgroups or Levi subalgebras.

There are versions for:


 * simply connected real Lie groups
 * algebraic Lie algebras over fields of characteristic zero
 * simply connected algebraic groups over fields of characteristic zero