I am coming from this question, which has not being completely answered but I think is very interesting.

In several works ([Chern], [Griffiths] and [Clelland]) the Maurer-Cartan form for $E(n)$ is worked out in the following manner. They consider maps from $G=E(n)$ to $\mathbb{R}^n$, $x, e_1,\ldots, e_n$, and express their differentials in terms of the frame in which we are. But for me that doesn't seem natural because is something very particular of this example: the frame itself can be described in terms of the objects it describe. I consider more natural the general approach: the group $E(n)$ can be seen like a matrix group of a special type, that one with elements of the form $$ \begin{pmatrix} A & v\\ 0 & 1\\ \end{pmatrix} $$ with $A\in O(n)$ and $v\in \mathbb{R}^n$. And now you only have to apply the formula for MC form for a matrix group, $\theta=g^{-1}dg$, obtaining the same 1-forms.

Question 1 (solved)

Is this true for every Lie group of this type? That is, whenever we have a group $G\approx \mathbb{R}^n \rtimes H$ it can be seen as a subgroup of $GL(n+1)$ as above (see this QA in MSE) and we can interpret the columns as vectors in the homogeneous space $G/H\approx \mathbb{R}^n$. Then, does the Maurer-Cartan form tell us the variation of these vectors expressed in the current frame?

Back to the case of $E(2)$, for simplicity. The MC form is $$ \theta=g^{-1}dg=\begin{pmatrix} 0&-d\theta&cos(\theta)da+sin(\theta)db\\ d\theta&0&-sin(\theta)da+cos(\theta)db\\ 0&0&0&\\ \end{pmatrix} $$ If we consider the basis of $\mathfrak{e}(2)$ given by $$ B=\left\{ \begin{pmatrix} 0&0&1\\ 0&0&0\\ 0&0&0 \end{pmatrix}, \begin{pmatrix} 0&0&0&\\ 0&0&1\\ 0&0&0&\\ \end{pmatrix}, \begin{pmatrix} 0&-1&0&\\ 1&0&0\\ 0&0&0&\\ \end{pmatrix} \right\}\equiv $$ $$ \equiv\{\partial_a|_e,\partial_b|_e,\partial_{\theta}|_e\} $$ we have $$ \theta=\mu_1 \otimes\partial_a|_e+\mu_2 \otimes\partial_b|_e +\mu_3\otimes \partial{\theta}|_e $$ with $$ \mu_1=cos(\theta)da+sin(\theta)db $$ $$ \mu_2=-sin(\theta)da+cos(\theta)db $$ $$ \mu_3=d\theta $$ In this case the Maurer-Cartan form has "two parts": $\mu_1, \mu_2$ on the one hand, and $\mu_3$ on the other hand. I think that $(\mu_1, \mu_2)$ corresponds to the canonical solder form and $\mu_3$ is the connection form of the Levi-Civita connection.

Question 2

Why is this the Levi-Civita connection? What relationship does it have (if any) with the group reduction of $GL(2)$ to $O(2)$ by means of the standard metric?

I have an intuition about some relation but I cannot grasp what it is... I know that the frame bundle for $\mathbb{R}^2$ is $\mathbb{R} \rtimes GL(2)$ and that the standard metric let us reduce the structure group of this principal bundle to $O(2)$...

**References**

[Chern]: Chapter 6 of S.S. Chern's book "Lectures on differential geometry"

[Griffiths]: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry

[Clelland]: From Frenet to Cartan: The Method of Moving Frames