Let $\omega$ be a 1-form and $X$ and $Y$ the vector fields on $M$, a smooth manifold. I know that there is a well-known identity:
which illustrates the relationship between exterior derivative and Lie bracket. The proof of this follows from taking co-ordinates of $X$ and $Y$ and running some computations, which is not difficult. However, I don't get the intuitive meaning behind this.
Is it simply meant to be seen as things like Leibniz rule, where the result, although computational, is quite fundamental that we are to take it as theoretical building-blocks? Indeed, this identity is key in showing that given a connection $\nabla$ with connection 1-form matrix $\Omega$ and curvature tensor $R$, $R(X,Y)(Z)=\nabla^2(Z)(X,Y)$(or the curvature matrix of $R$ is indeed $d\Omega-\Omega \wedge \Omega$). Since Lie bracket $[X,Y]$ measures(in some sense) how far the vector fields $X$ and $Y$ are off from forming local co-ordinates, we can think of the term $\omega[X,Y]$ as making 'adjustments', but I wonder whether there are better interpretations of this identity. The cases where $X=\partial/\partial x^i$ and $Y=\partial/\partial x^j$ do return what is expected- that is, $d\omega(x,y)=\partial \omega^j/\partial x^i-\partial \omega^i/\partial x^j$, but it doesn't really contribute to explaining the $\omega[X,Y]$ term.