  In a previous work, we proved strong convergence with order $1/2$ of the
Ninomiya-Victoir scheme $X^{NV,\eta}$ with time step $T/N$ to the solution $X$
of the limiting SDE. In this paper we check that the normalized error defined
by $\sqrt{N}\left(X - X^{NV,\eta}\right)$ converges to an affine SDE with
source terms involving the Lie brackets between the Brownian vector fields. The
limit does not depend on the Rademacher random variables $\eta$. This result
can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central
limit theorem of Lindeberg Feller type, derived by M. Ben Alaya and A. Kebaier
for the multilevel Monte Carlo estimator based on the Euler scheme. When the
Brownian vector fields commute, the limit vanishes. This suggests that the rate
of convergence is greater than $1/2$ in this case and we actually prove strong
convergence with order $1$.
