A continuous local martingale $M$ is a Brownian motion if and only if $[M]_t=t$ for every $t\ge 0$. Or equivalently if and only if $M_t^2-t$ is a continuous local martingale.
How do we prove these two conditions are equivalent? This is to show that $[M]_t=t$ for every $t\ge 0$ is equivalent to $M_t^2-t$ is a continuous local martingale.
My work: if $[M]_t=t$, we know that $M_t^2-[M]_t=M_t^2-t$ is a continuous local martingale. But how do show that the converse statement?