  In a previous work, we proved strong convergence with order $1$ of the
Ninomiya-Victoir scheme $X^{NV}$ with time step $T/N$ to the solution $X$ of
the limiting SDE when the Brownian vector fields commute. In this paper, we
prove that the normalized error process $N \left(X - X^{NV}\right)$ converges
to an affine SDE with source terms involving the Lie brackets between the
Brownian vector fields and the drift vector field. This result ensures that the
strong convergence rate is actually $1$ when the Brownian vector fields
commute, but at least one of them does not commute with the drift vector field.
When all the vector fields commute the limit vanishes. Our result is consistent
with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.
