== Maurer-Cartan equations == 

 I believe there is a mistake in the Maurer-Cartan equations as they are mentioned in the section https://en.wikipedia.org/wiki/Affine_connection#Affine_frames_and_the_flat_affine_connection. The minus signs should be plus signs. Starting from 
 \begin{align} 
 \text{d}\pi &= \theta^i \varepsilon_i\\ 
 \text{d}\varepsilon_i &= {\omega^k}_i \varepsilon_k 
 \end{align} 

  
 and computing as indicated on the aforementioned article, if is found that 

  

 as well as 

  

 Using the linear independence of the functions  yields the Maurer-Cartan equations but with plus signs between the terms instead of the minus signs on the article.