What is the adjoint orbit of element of lie algebra?

Question

I was studying paper by GRANT CAIRNS about geodesic basis. https://arxiv.org/pdf/1312.2186.pdf There I was about adjoint orbit of element of lie algebra. I only know adjoint representation of lie algebra. but if
Let Z is a element of a lie algebra g. how do we define the adjoint orbit of Z ?

Answer 1

Take any connected Lie group $G$ with Lie algebra $\mathfrak{g}$. (This exists by Lie's third theorem.) The adjoint orbit of an element $Z\in\mathfrak{g}$ is defined by $$G\cdot Z=\{\mathrm{Ad}_gZ:g\in G\},$$ where $\mathrm{Ad}_g$ is the differential at $1\in G$ of the conjugation map $C_g:G\to G$, $h\mapsto ghg^{-1}$.

It is not difficult to show that if $G_1$ and $G_2$ are two connected Lie groups with Lie algebra $\mathfrak{g}$, then $G_1\cdot Z=G_2\cdot Z$ for all $Z\in\mathfrak{g}$. So this is well-defined, independently of the chosen Lie group $G$. The idea of the proof is to show that $\tilde{G}_1\cdot Z=G_1\cdot Z$, where $\tilde{G}_1$ is the universal covering group of $G_1$. Then, use that $\tilde{G}_1=\tilde{G}_2$, which follows since they have the same Lie algebra.