= Solution
Let $\tau_n$ localize $M$. The <Itô isometry> gives
$$
\mathbb E M_{t\wedge\tau_n}^2
=\mathbb E[M]_{t\wedge\tau_n}
\leq\mathbb E[M]_t.
$$
Thus the stopped variables are bounded in $L^2$, and localization plus weak compactness shows that $M$ is a true martingale. The <Itô formula> applied to $f(x)=x^2$ gives
$$
M_t^2-[M]_t=2\int_0^tM_s\,dM_s.
$$
After localization this is a martingale; the <Burkholder-Davis-Gundy inequalities> and $\mathbb E[M]_t<\infty$ supply the required local integrability, so it is a true martingale.
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