Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals.)

Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra ${\displaystyle {\mathfrak {g}}}$, if nonzero, the following conditions are equivalent:

• ${\displaystyle {\mathfrak {g}}}$ is semisimple;
• the Killing form ${\displaystyle \kappa (x,y)=\operatorname {tr} (\operatorname {ad} (x)\operatorname {ad} (y))}$ is non-degenerate;
• ${\displaystyle {\mathfrak {g}}}$ has no non-zero abelian ideals;
• ${\displaystyle {\mathfrak {g}}}$ has no non-zero solvable ideals;
• the radical (maximal solvable ideal) of ${\displaystyle {\mathfrak {g}}}$ is zero.