I am having troubles in understanding the modern definition of moving frames method. 

Classically, the idea of moving frames is "to express the variation in terms of the moving frame itiself". This is clear if we have a matrix Lie group $G$, like $O(n), GL(n)$...

But if $G$ is not a matrix group, we _cannot_ express the variation in terms of the moving frame. At this point are introduced Lie algebras, Maurer-Cartan forms and all things that I understand but I cannot figure out where they come from. 

If $G$ is a matrix Lie group, the Maurer-Cartan form is the usual that I know and so is the Maurer-Cartan equation and that's ok. My question is: why is this the correct generalization of moving frame method to "abstract" Lie group? What's the intuition behind this? Who does this generalization?

EDIT

In particular I am asking what's the intuition behind these generalizations:

1. _From_ the matrix Maurer-Cartan form $\omega = g^{-1}dg$ _to_ the "the unique left-invariant $\mathfrak{g}$-valued 1-form on $G$ such that $\omega|_e : T_eG \to \mathfrak{g}$ is the identity.
2. _From_ the matrix Maurer-Cartan equation $\text{d}\omega = -\omega\wedge\omega$ _to_ the equation $\text{d}\omega = -\frac{1}{2}[\omega, \omega]$.
3. Why when need the Lie bracket if before we don't?

EDIT II

I tought this answers for 1. In "classic" moving frames method, $G$ acts on every $T_gG \cong \mathbb{R}^{n, n}$ because $G$ is a matrix group. But this is true _also_ for an _abstract_ Lie group $G$. In fact we have the left action of $G$ on $TG$ given by

\begin{align}
\cdot: G \times TG &\to TG \\
(g, v) &\mapsto \text{d}L_g(v)
\end{align}

So if we have a moving frame $f: M \to G$ we also have that $df = f \cdot \omega_f$ with $\omega_f: TM \to TG$. It's easy to see that really we have that $\omega_f: TM \to T_eG$ and also that $\omega_f = f^*\omega$ with $\omega: TG \to T_eG$. This is the Maurer-Cartan form, but _this_ approach is for me the same as Cartan's; for me defining the Maurer-Cartan form as "the unique..." breaks the intuition behind the method. 

In these terms I would reformulate question 2. Classically, we start from $dg = g\omega$ and we exterior-differentiate to get $0 = \text{d}g\wedge\omega + g\text{d}\omega = g(\omega \wedge \omega + \text{d}\omega)$ and so the Maurer-Cartan equation. My question is: can we reply this approach starting from $df = f \cdot \omega_f$? For me makes sense that we have to use Lie bracket since is the analogous of $\text{d}^2$ in the antisymmetrization process: but how?