 Solution

The $d$-dimensional Lévy characterization of Brownian motion says that a continuous local martingale $X$ with $X_0=0$ is a standard $d$-dimensional Brownian motion exactly whenfor all $i,j$ and $t$.

One direction follows directly from independent Gaussian increments. Conversely, fix $\theta \in {\mathbb{R}}^d$. Applying Itô formula and the bracket assumption shows thatis a complex local martingale. After stopping $X$ on leaving large balls it is bounded, so optional sampling and then dominated convergence give, for $s<t$,This is the characteristic function of $N(0,(t-s)I_d)$ and is deterministic. Thus each increment is Gaussian with the required covariance and independent of the past. Together with continuity, these are precisely the defining properties of standard $d$-dimensional Brownian motion.