If $E$ is a real bundle and $\nabla$ a connection, each $e \in E$ gives us $\omega_e := \nabla_e \in Hom(T,E)$.
The cartan-maurer equation is that using the connection twice we get
$$\theta_e = d\omega_e + \frac{[\omega_e,\omega_e]}{2}$$
I am trying to interpret what the right side even means.
If we choose a trivialization of $E$ this says that if (using Einstein summation to show off) $\nabla e_j = \omega_j^i \otimes e_i$ then
$$\theta_e (e_j) = [d\omega_j^i + \omega^i_s \wedge \omega_{s,j})]\otimes e_i $$
so syntactically we see the $d$ and the commutator, but only syntactically in local coordinates.
We can try to make an intrinsic interpertation via saying; for a fixed $v \in T$, we have $\nabla_v: E \to E$ and we can really take commutators of such maps. That's misleading though, because the map is not linear, and when we compare the coordinate computation to this statement we are pretending that it is linear.
Conversely, we can try to fix $e \in E$ and obtain $\nabla_*(e) \in Hom(T, E)$, but there is no meaning to a commutator of such maps.
Thus my question is how should I be interpreting the commutator in cartan's equation; is it really only very coordinate dependent syntactic statement or is there a way to actually give it meaning?
Via the group principle bundle we get an actual intrinsic definition, but can we translate it?
EDIT: As Quaere Verum points out, these equations are known as Cartan's structure equations and not the Maurer-Cartan equations. They're closely related, which is why I prefer to just call both the Maurer-Cartan equations. For example, if you restrict the Maurer-Cartan equations on the group of rigid motions of $\mathbb{R}^n$ to the frame bundle of a submanifold of $\mathbb{R}^n$, then you get the structure equations of the submanifold.
Still, it's better to minimize confusion here.
Also, I often start from scratch and write long answers to minimize confusion about the notation and definitions. Here, it is tempting to try to define the connection $1$-form as a bundle-valued $1$-form or something like that. However, it really is a matrix-valued $1$-form with indices.
But this Given a local section $s: O \rightarrow E$, where $O\subset M$ is open, there is, for each $p \in O$, a linear map \begin{align*} \nabla s: T_pM &\rightarrow E_p\\ X &\mapsto \nabla_Xs. \end{align*}
Let $\mathcal{F}$ denote the frame bundle of $E$. In other words, for each $p \in M$, $\mathcal{F}_p$ is the set of all bases of $E_p$.
Given a frame $e =(e_1, \dots, e_m) \in \mathcal{F}_p$ at $p$, you can define a matrix of $1$-tensors $\omega^i_j \in T^*_pM$, $1 \le i,j \le m$ at $p$ as follows: Extend the frame $e = (e_1, \dots, e_m)$ to a frame of sections $f = (f_1, \dots, f_m)$ on a neighborhood of $p$. Then there is a matrix of $1$-forms $\overline\omega^i_j$, $1 \le i,j \le m$, such that $$ \nabla f_j = f_i\overline\omega^i_j. $$ Let $\pi: \mathcal{F} \rightarrow M$ be the natural projection map and let $$ \omega^i_j = \pi^*_e\overline\omega^i_j \in T_e^*\mathcal{F}. $$ You can check that $\omega^i_j$ is independent of the section $f$ used, as long as $f(p)=e$. This defines a matrix of global $1$-forms $\omega^i_j$ on $\mathcal{F}$. You can now check that the $1$-forms $\omega^i_j$ define for each $e \in \mathcal{F}$ a linear bundle map \begin{align*} T_*\mathcal{F}\otimes \pi^*E &\rightarrow \pi^*E\\ X\otimes v &\mapsto e_i\langle\omega^i_j,X\rangle v^j, \end{align*} where $v = e_jv^j$.
Let $$ \Omega^i_j = d\omega^i_j + \omega^i_k\wedge\omega^k_j. $$ You can check that it defines a bundle map \begin{align*} T_*\mathcal{F}\otimes T_*\mathcal{F}\otimes \pi^*E &\rightarrow \pi^*E\\ X\otimes Y \otimes v &\mapsto e_i\langle \Omega^i_j,X\otimes Y\rangle v^j. \end{align*}
Now the observation is \begin{align*} \langle \omega^i_k\wedge\omega^k_j, X\otimes Y\rangle &= \langle \omega^i_k,X\rangle\langle\omega^k_j,Y\rangle - \langle \omega^i_k,Y\rangle\langle\omega^k_j,X\rangle\\ &= [\langle\omega,X\rangle, \langle \omega,Y\rangle]^i_j. \end{align*} On the other hand, we can define $[\omega,\omega]$ to be the matrix-valued $2$-form where $$ \langle[\omega,\omega],X\otimes Y\rangle = [\langle\omega,X\rangle,\langle\omega,Y\rangle] - [\langle\omega,Y\rangle,\langle\omega,X\rangle] = 2[\langle\omega,X\rangle,\langle\omega,Y\rangle]. $$ The index-free version of Cartan's structural equations is therefore $$ \Omega = d\omega + \omega\wedge\omega = d\omega + \frac{1}{2}[\omega,\omega]. $$