= 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 when
$$
[X^i,X^j]_t=\delta_{ij}t
$$
for 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 that
$$
Z_t=\exp\left(i\theta\mathbin\cdot X_t+\frac12|\theta|^2t\right)
$$
is a complex local martingale. After stopping $X$ on leaving large balls it is bounded, so optional sampling and then dominated convergence give, for $s<t$,
$$
\mathbb E\left[e^{i\theta\cdot(X_t-X_s)}\mid\mathcal F_s\right]
=e^{-\frac12|\theta|^2(t-s)}.
$$
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.
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