Example: basis of left invariant vector fields and MC form

The case of $E(2)$

xournal 192

Consider the euclidean group $E(2)={\mathbb{R}}^2⋊SO(2)$. It can be realized as a matrix group with a local chart given by

for $a,b,\theta \in \mathbb{R}\times \mathbb{R}\times (-\pi ,\pi )=:U$.

I am looking for a basis of the left invariant vector fields. In the parameter space $U$ I take a basis of $T_{e}U$, $\{{\partial }_a,{\partial }_b,{\partial }_{\theta }\}$ and apply $d(L_g{)}_e$ to all of them, being $g=(a_0,b_0,{\theta }_0)$ and

the left translation by $g$.

This way I obtain the basis of left invariants vector fields

But now I want to see this basis in $GL(3)$.
Because even if this example is fairly easy to work with it in parameter space, with a group with a more complicated "product" can be difficult to deal. In $GL(3)$, left translation $L_g$ takes the shape of left multiplication by the matrix $\phi (g)$, satisfying the commutative diagram

A basis for $\mathfrak{g}=T_{e}E(2)\subset T_{e}GL(3)$ is $B=\{d{\phi }_e({\partial }_a),d{\phi }_e({\partial }_b),d{\phi }_e({\partial }_{\theta })\}$, that is,

Multiplying by the left with

we obtain the left invariant vector fields in $TE(2)\subset TGL(3)$

It can be checked that they are precisely $\{{\phi }_{\ast }(e_i)\}$.

By the way, given any left invariant vector field, its coordinates in the basis above are the Maurer-Cartan forms. Consider a vector field $V=v_1{\partial }_a+v_2{\partial }_b+v_3{\partial }_{\theta }\in T_{g}U$. It yields a vector field $d\phi (V)\in GL(3)$:

We can take $da,db,d\theta$ instead of $v_1,v_2,v_3$ and then

The $\mathfrak{g}$-valued Maurer-Cartan form is, according to the formula MC form for a matrix group

but, what are the real valued Maurer-Cartan forms? Since we are working with basis $B$ above, we have

Question
Coming from this question.
I find the approach of Chern, @griffiths1974cartan and Clelland very misleading. They consider maps from $G=E(n)$ to ${\mathbb{R}}^n$, $x,e_1,\dots ,e_n$, 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 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

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.

Is this true for every Lie group of this type? That is, whenever we have a group $G\approx {\mathbb{R}}^n⋊H$ 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

If we consider the basis of $\mathfrak{e}(2)$ given by

we have

with

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 Levi-Civita connection.

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? See Euclidean plane.