I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.


I can see in many places the definition of the Lie derivative with respect to a vector field, and I also understand that an element of a Lie algebra $\theta$ can be considered as a left (or right) invariant vector field whose value at the identity is $\theta$.


But I would really like a sweet, short reference that combines both of these into one definition, or at least states this explicitly that this can be done.


What I really want is something accessible to engineers with a weak math background. As an example, think of $SO(n)$ as embedded in $\mathbb R^{n^2}$, and suppose $f:SO(n) \to \mathbb R$. Then the 'Lie derivative' of $f$ in the direction of a an anti-symmetric matrix $A$ at $Q$ is
$$ \mathcal L\_A(Q) = \frac d{dt} f\big(Q(I+tA)\big) \Bigg|\_{t=0} .$$



Lie derivative on Lie group in the direction of an element of Lie algebra
What about the formula $(D\_X\varphi)(g)=\frac{\rm d}{{\rm d}t}\Bigl\vert\_{t=0}\varphi(g\exp\_G(tX))$ for $\varphi\in C^\infty(G)$, $g\in G$, and $X\in \mathfrak g$, where $G$ is a Lie group with its Lie algebra $\mathfrak g$? See for instance Eq. (5) in Ch. II of the book by S.Helgason, "Differential geometry, Lie groups, and symmetric spaces".


