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The radial martingale $H=\int XdX+YdY$ and the planar Brownian stochastic area $Z=\int YdX-XdY$ have common bracket $A=\int (X^2+Y^2)ds$ and zero cross variation, because their integrand vectors $(X,Y)$ and $(Y,-X)$ are orthogonal. The Itô formula also gives $X_t^2+Y_t^2=2H_t+2t$.