 Additivity of independently driven squared Bessel processes

For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.