Theorem of Levi

Levi's Theorem states that any complex Lie algebra can be broken down into a direct sum of two parts: a semisimple part, which has a highly symmetrical structure, and a solvable part, which can be thought of as more "flexible" or "deformable." This decomposition helps mathematicians understand the algebra's overall structure by analyzing these two components separately. It’s similar to identifying a sturdy core and a flexible outer layer in a complex system, making the study and classification of such algebras more manageable and systematic.