= Solution
The scalar <Lévy characterization of Brownian motion> states that an <adapted process> $X$ starting at zero is a <Brownian motion> if and only if it is a <continuous local martingale> with
$$
\boxed{\langle X\rangle_t=t.}
$$
For necessity, the centered independent increments make a <Brownian motion> a <martingale>. Their conditional second moments show that $X_t^2-t$ is also a <martingale>. The defining uniqueness of the <quadratic variation> compensator gives $\langle X\rangle_t=t$.
For sufficiency, fix $\theta\in\mathbb R$ and apply the <Itô formula> to
$$
Z_t=\exp\left(i\theta X_t+\frac{\theta^2t}{2}\right).
$$
The time drift cancels the second-order Itô term, leaving $dZ_t=i\theta Z_t\,dX_t$. Its real and imaginary parts are <local martingales>. On any deterministic interval $[0,T]$, $|Z_t|=e^{\theta^2t/2}\leq e^{\theta^2T/2}$, so the <bounded local martingale criterion> makes them true <martingales>. Therefore
$$
\mathbb E[e^{i\theta(X_t-X_s)}\mid\mathcal F_s]
=e^{-\theta^2(t-s)/2}.
$$
Part (b) proves the Brownian property. This also supplies a proof of <Lévy's characterization of Brownian motion> through conditional Fourier transforms.
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