Characterization of differentiability via Lie derivatives

Let us consider what happens on $\mathbb{R}^n$. We denote by $X_1,\dots,X_n$ the standard vector fields. Then it is well known that $f$ is $C^k$ iff for any $i_1,\dots,i_k$, the directional derivatives $X_{i_1}.(X_{i_2}(\dots.(X_{i_k}.f)\dots)$ exist and are continuous.

Now take any other vector fields $Y_1,\dots,Y_n$ giving a basis of $T_x \mathbb{R}^n$ for $x$ in a neighborhood of $x_0$. Then you can write $Y_i = \lambda_i^j X_j$, with smooth $\lambda_i^j$. This shows that if $f$ is $C^k$ in a neighborhood of $x_0$ then the directional derivatives $Y_{i_1}.(Y_{i_2}(\dots.(Y_{i_k}.f)\dots)$ exist and are continuous. Reversing the argument, this shows that in order to show that a map is $C^k$, you can compute directional derivatives for any vector fields giving a basis of the tangent space.

Now, for your Lie group, $L_X f$ is just the directional derivative of $f$ in the direction of the left-invariant vector field associated to $X$. I think this proves what you want. Or am I missing something ?