The formula defining the Riemann Curvature Tensor $R$ is:
$$R(u,v)w = \nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w$$
The most common geometric explanation I see for this formula is "parallel transporting $w$ around a small parallelogram defined by $u$ and $v$, and seeing how much $w$ 'twists' around". However, if this is the case, I don't understand what the $\nabla_{[u,v]}w$ term in the formula represents.
When deriving this formula, most sources (like this one on page 8) assume that $u$ and $v$ are coordinate vector fields, and therefore take $[u,v]=0$, sending the third term in the above formula to zero during the derivation.
What's the geometric meaning of the third term with the lie bracket? Are there any derivations of the Riemann Tensor which include it?
I know when the Lie bracket is non-zero, flow curves don't form a perfect parallelogram, and the Lie bracket vector "closes the gap", in a sense. Is this relevant?
(I also see many sources saying the Riemann tensor is the "commutator" of the covariant derivative. However, this isn't strictly true if we include the 3rd term.)