Adjoint representation I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't understand this terminology. For sure, $Ad$ does not enables us to represent elements in the Lie group and $ad$ does not enable us in general to represent every element in $\mathfrak{g}$, so where does the name come from?

The term representation comes from representation theory.  In particular, Ad gives us a group representation, and ad gives us a Lie-algebra representation.
In particular: given a Lie group $G$ and $A,B \in G$, $\operatorname{Ad}$ is a map from $G$ to $GL(\mathfrak{g})$ such that
$$
\operatorname{Ad}_A \operatorname{Ad}_B = \operatorname{Ad}_{A B}
$$
Similarly: given a Lie algebra $\mathfrak g$ and $X,Y \in G$, $\operatorname{ad}$ is a map from $\mathfrak g$ to $\mathfrak{gl(g)}$ such that
$$
[\operatorname{ad}_X,\operatorname{ad}_Y] = \operatorname{ad}_{[X,Y]}
$$
While these need not be full or faithful representations, they are representations nevertheless.
