Adjoint action of Lie group on Lie algebra

Definition for a real Lie group
Suppose $$G$$ is a real Lie group and $$\mathfrak{g}$$ is its Lie algebra. The adjoint action of $$G$$ on $$\mathfrak{g}$$ is a homomorphism of groups from $$G$$ to the automorphism group of $$\mathfrak{g}$$, i.e., a homomorphism:

$$\operatorname{Ad}: G \to \operatorname{Aut}\mathfrak{g}$$

The map is defined as follows: for $$g \in G$$, $$\operatorname{Ad}(g)$$ evaluated at $$x \in \mathfrak{g}$$ is defined as follows.


 * 1) First, find the unique one-parameter group $$\gamma:\R \to G$$ such that $$\gamma'(0) = x$$.
 * 2) Consider the new one-parameter group $$\beta = c_g \circ \gamma$$ where $$c_g$$ is the inner automorphism defined as conjugation by $$g$$. In other words, we define $$\beta(t) = g\gamma(t)g^{-1}$$.
 * 3) Now, take the tangent vector $$\beta'(0)$$. This is the desired answer.

Definition for Lie groups over other topological fields
This is similar to the definition for a real Lie group.

Definition for a linear Lie group
Suppose $$G$$ is a linear Lie group over a topological field $$K$$, i.e., a Lie group with an embedding as a closed subgroup of the general linear group $$GL(n,K)$$ (where the closed is relative to the topology). Suppose $$\mathfrak{g}$$ is the Lie algebra of $$G$$. The adjoint action of $$G$$ on $$\mathfrak{g}$$ is a homomorphism of groups from $$G$$ to the automorphism group of $$\mathfrak{g}$$, i.e., a homomorphism:

$$\operatorname{Ad}: G \to \operatorname{Aut}\mathfrak{g}$$

The map is defined as follows: for $$g \in G$$ and $$x \in \mathfrak{g}$$, we define

$$\operatorname{Ad}(g)(x) := gxg^{-1}$$

where the multiplication on the right side is matrix multiplication.

Definition for a Lazard Lie group
If $$G$$ is a Lazard Lie group, its Lazard Lie ring can be identified with $$G$$ as a set, with the Lie ring operations defined in terms of the group operations of $$G$$. The adjoint action of the group $$G$$ on itself as a Lie ring is simply the group action on itself by conjugation, now viewed as an action on itself as a Lie ring.