nLab
simple Lie group

Definition

A simple Lie group is a non-abelian connected Lie group with no nontrivial connected normal subgroups.

Remark

This is not the same thing as a simple object in LieGrp (or even in $Conn Lie Grp$). However, the Lie algebra associated to a simple Lie group is always a simple Lie algebra (although that definition also requires a non-abelian clause).

Classification

The classification of simple Lie groups consists of four infinite series – the classical Lie groups – and five separate cases – the exceptional Lie groups.

See Wikipedia's list of simple Lie groups.

See also at ADE classification