= Solution
The <Lévy characterization of multidimensional Brownian motion> says that a continuous adapted $\mathbb R^d$-valued process $Z$, with $Z_0=0$, is standard <Brownian motion> relative to the given <filtration> if and only if its coordinates are <local martingales> and
$$
\boxed{[Z^i,Z^j]_t=\delta_{ij}t\qquad(1\le i,j\le d).}
$$
For <Brownian motion>, independent centered Gaussian increments give the <martingale> property of each coordinate and of $Z_t^iZ_t^j-\delta_{ij}t$. The defining uniqueness of <quadratic covariation> consequently gives the bracket condition.
Conversely, suppose the <local martingale> and bracket conditions hold. For $\theta\in\mathbb R^d$, the <Itô formula> gives
$$
E_t=\exp\left(i\theta\cdot Z_t+\tfrac12|\theta|^2t\right),\qquad dE_t=iE_t\,\theta\cdot dZ_t,
$$
since the bracket term cancels the time derivative. On $[0,T]$, $|E_t|\le e^{|\theta|^2T/2}$, so its real and imaginary parts are bounded <local martingales> and hence true <martingales>. Thus, for $s\le t$,
$$
\mathbb E\left[e^{i\theta\cdot(Z_t-Z_s)}\mid\mathcal F_s\right]=e^{-\frac12|\theta|^2(t-s)}.
$$
This is the <characteristic function> of $N(0,(t-s)I_d)$ and is independent of $\mathcal F_s$. Uniqueness of <characteristic functions> shows that $Z_t-Z_s$ has that <Gaussian distribution> and is independent of the past. Iterating over ordered times gives independent Gaussian increments; together with continuous paths and $Z_0=0$, these are precisely the defining properties of standard <Brownian motion>.
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