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For a detailed discussion, see What’s so special about the adjoint representation of a Lie group? by J. Schwichtenberg
A representation is a map that maps each element of the set of abstract groups element to a matrix that acts on a vector space. A confusing point here is: If we can study the representation of any group on any vector space, where should we start?
Luckily there comes exactly only distinguished vector space automatically with each Lie group: the Lie algebra of the group!
The representation of each group where it acts on its own Lie algebra is called the adjoint representation.