  We re-examine the celebrated Doob--McKean identity that identifies a
conditioned one-dimensional Brownian motion as the radial part of a
3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now
in the analogous setting of isotropic $\alpha$-stable processes. We find a
natural analogue that matches the Brownian setting, with the role of the
Brownian motion replaced by that of the isotropic $\alpha$-stable process,
providing one interprets the components of the original identity in the right
way.
