Semisimple Lie group

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras ${\displaystyle {\mathfrak {g}}}$ whose only ideals are {0} and ${\displaystyle {\mathfrak {g}}}$ itself.

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

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