Definition 7

Let $g$ be a Lie algebra and $r$ its radical. A Levi subalgebra of $g$ is any subalgebra of $g$ supplementary to $r$.

A Levi subalgebra is isomorphic to $g/r$ and hence is semi-simple. As a semi-simple subalgebra has only 0 in common with $r$, every semi-simple subalgebra $h$ such that $g = r + h$ is a Levi subalgebra; consequently the image of a Levi subalgebra under a surjective homomorphism is a Levi subalgebra.