ABSTRACT
Sophus Lie shows how the generators can actually be defined in the abstract group without mentioning representations at all. The Lie algebra is completely determined by the structure constants. Each group representation gives a representation of the algebra in an obvious way, and the structure constants are the same for all representations because they are fixed just by the group multiplication law and smoothness. The structure constants themselves generate a representation of the algebra called the adjoint representation. An invariant subalgebra is some set of generators which goes into itself under commutation with any element of the algebra. Algebras without Abelian invariant subalgebras are called semisimple.
