I want to show that: if for all $\lambda \in \mathbb{R}$ the process $\left(\exp\left(\lambda X_t-\frac{\lambda ^2}{2}t\right)\right)_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process $X$ is a $\mathcal{F}^X$-Brownian motion
I think it is enough to show that $X_t$ is a continuous local martingale and that for all $t$: $[X]_t=t$, because then by Lévy's Characterization $X$ has to be a Brownian motion.
I tried using Ito's Lemma but couldn't get there.