A holonomic or coordinate basis for a differentiable manifold is a set of basis vector fields $\{e_k\}$ such that some coordinate system $\{x_k\}$ exists such that $e_k=\partial/\partial x_k$.
A local condition for a basis to be holonomic is $[e_i,e_j]=0$. Can one always find a set of basis vector fields on a manifold which satisfy $[e_i,e_j]=0$ locally? Does this hold for all manifolds, including Lie groups?