= Semisimple Lie group
= Semisimple group
{synonym}
A real or complex <Lie group> is semisimple when its <Lie algebra> is semisimple. For a real group, the <Killing form> then defines a nondegenerate <bi-invariant pseudo-Riemannian metric>. Compact semisimple groups have negative-definite <Killing form>; noncompact examples such as $SL(2,\mathbb R)$ have indefinite <Killing form>. A disconnected group can also have a <semisimple Lie algebra>, so connectedness is a separate hypothesis when integrating infinitesimal actions.
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