Maurer-Cartan form on a matrix Lie group
In case $G$ is a matrix Lie group, i.e., $G\subset GL(n)$ then the (left-invariant) Maurer-Cartan form can be computed with the following formula
$$ \theta=g^{-1} dg $$This is the expression appearing everywhere, but I think the correct expression for $\theta$ should be, for $p\in G$ and $V\in T_pG$
$$ \theta_p (V)=dg_e^{-1}(g(p)^{-1} dg_p(V)) $$
But what is $dg$? Following @ivey2016cartan definition 1.6.1, $g:G\to M_{n\times n}$ is a parametrization (embedding) of the group as a submanifold of $M_{n\times n}$ but the name $g$ is a bit misleading. For example, consider for $S\subset \mathbb{R}^2$ the embedding
$$ \iota: S \to \mathbb{R}^3 $$given by $\iota(a,b)=(a^2+b,-b,a+b)$. The differential $d\iota$ is a map
$$ d\iota: TS \to T \mathbb{R}^3 $$and we know that for a vector $(v_1,v_2)\in T_p S$ the image is computed as
$$ d\iota_p(v)=\begin{pmatrix} 2a& 1 \\ 0&-1\\ 1&1\\ \end{pmatrix}\cdot \begin{pmatrix} v_1\\ v_2 \end{pmatrix}= \begin{pmatrix} 2av_1+v_2\\ -v_2\\ v_1+v_2 \end{pmatrix}. $$If we use the natural identification $T_{\iota(p)} \mathbb{R}^3\approx \mathbb{R}^3$ we can think of $d\iota$ as the $\mathbb{R}^3$-valued differential 1-form
$$ d\iota=(2ada+db,-db,da+db) $$In the case of $g:G\to M_{n\times n}$, think in $G$ as the parameter space and $M_{n\times n}$ a fancy way of writing $\mathbb{R}^{n^2}$. So $dg$ is a $M_{n\times n}$-valued differential form in $G$. This way, the expression $\theta=g^{-1}dg$ is also a $M_{n\times n}$-valued differential form in $G$.
Examples
I have worked examples in xournal 188 (afin group) and xournal 189 (euclidean group).
Interpretation
In xournal 189 there is a reflection on how do we understand the Maurer-Cartan form when the group is interpreted as a set of frames. If $G$ is the set of frames or $G$-descriptions of a homogeneous space $X$ (see this), then for a frame $f\in G$, a vector $V\in T_f G$ can be thought as an arrow connecting the frame $f$ with a "nearby" frame $\bar{f}$, i.e., something like $V\approx \bar{f}-f \approx \delta f$. That is, $V$ is a infinitesimal change of the frame $p$. In this context, $\theta_f(V)$ is nothing but how this infinitesimal change can be described from the own frame $f$.
The same in other words (following ideas of the end of this): given a frame $g\in G$, we can consider a nearby frame $\bar{g}\approx g+dg$. To express this "frame" in the frame $g$ we take
$$ g^{-1}(g+dg)=I+g^{-1}dg. $$So the Maurer-Cartan form $g^{-1}dg$ is the "infinitesimal displacement" of the frame, but described from the point of view of the chosen frame.
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Author of the notes: Antonio J. Pan-Collantes
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