The tangent space of $GL(n,\mathbb R)$ in the unit element is $Mat(n,\mathbb R)$ and its elements correspond to left-invariant vector fields on $GL(n,\mathbb R)$.
It is well-known that under this correspondence the commutator of matrices corresponds to the commutator of vector fields.
This can be shown by a direct computation which, though not difficult, is somewhat nasty because of the many indices you‘ve got to use.
My question: is there some conceptual argument explaining why commutators of matrices and vector fields are the same?