 Common-clock Brownian representation of radius and area

Jointly time-change the radial and area martingales by the inverse of $A=\int R^{2}ds$. The bracket matrix becomes $u{I}_2$, so the Lévy characterization of multidimensional Brownian motion gives two independent Brownian coordinates. The clock is strictly increasing; if it were finite at infinity, the radial martingale would converge and $R_t^2=2t+O(1)$ would force its integral to diverge. Thus $R_t^2=2W_{A(t)}+2t$ and $Z_t=B_{A(t)}$ with $W,B$ independent.