= Common-clock time change of orthogonal local martingales
{title2=$[M^i,M^j]_t=\delta_{ij}A_t\Longrightarrow(M^i_{\tau_r}-M^i_0)_i\text{ is Brownian}$}
If a continuous vector local martingale has bracket matrix $A_tI$ with $A_\infty=\infty$, inverse-clock time change gives bracket matrix $rI$. The <Lévy characterization of multidimensional Brownian motion> then gives Brownian motion in the common time-changed filtration. A continuous local martingale is constant on flat intervals of its bracket, so inverses using $A_t\geq r$ and $A_t>r$ give the same martingale values. The filtration at the earlier inverse is smaller; adaptation and independence from the larger past transfer the Brownian property to it. Nonzero initial values are retained by adding the initial vector back.
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