Matrix commutator from a logarithm chart ($\log (e^{tX}{e}^{tY}{e}^{-tX}{e}^{-tY})=t^2[X,Y]+O(t^3)$)

If the matrix logarithm identifies a neighborhood of the identity in a Matrix Lie group with a neighborhood of zero in its tangent subspace, the logarithm of the displayed group commutator belongs to that subspace. Divide it by $t^2$ and take the limit. Finite-dimensional linear subspaces are closed, proving closure under the matrix commutator without attempting to give a nonsmooth square-root reparametrization a velocity.