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 $$ \mathcal L_A(Q) = \frac d{dt} f\big(Q(I+tA)\big) \Bigg|_{t=0} .$$