A simple Lie algebra is non-abelian Lie algebra with no nontrivial ideals. A semisimple Lie algebra is a Lie algebra which is the direct sum of simple Lie algebras. This tag is for questions about semisimple Lie algebras, including their classification and correspondent to root systems and Dynkin diagrams.
A simple Lie algebra is non-abelian Lie algebra $\mathfrak{g}$ whose only ideals are $\{0\}$ and $\mathfrak{g}$. A semisimple Lie algebra is a Lie algebra which is the direct sum of simple Lie algebras. Every finite dimensional Lie algebra can be decomposed as the semidirect product of a solvable ideal and a semisimple subalgebra (this is the Levi decomposition).
Semisimple Lie algebras over an algebraically closed field are completely classified by their root systems) which, in turn, correspond to a collection of Dynkin diagrams. This, together with the Levi decomposition, make semisimple Lie algebras objects of interest in representation theory.
