# Integration of Maurer-Cartan form

Let $$G$$ be a Lie group with Lie algebra $$g$$. As it is well known the Maurer-Cartan form $$ω:TG\rightarrow g$$ transports any vector $$X\in T_{x}G$$ to the start $$l_{x^{-1}*}(X)\in g$$, $$l_{x^{-1}}$$ denoting the left translation. Let $$σ:[0,1]\rightarrow G$$ a smooth path on $$G$$. It there a way to define path integration on $$G$$ such that $$\int_{σ}{ω}=σ(1)σ(0)^{-1}$$?