  We intrinsically characterize separability of the Dirac equation in
Kerr-NUT-(A)dS spacetimes in all dimensions. Namely, we explicitly demonstrate
that in such spacetimes there exists a complete set of first-order mutually
commuting operators, one of which is the Dirac operator, that allows for common
eigenfunctions which can be found in a separated form and correspond precisely
to the general solution of the Dirac equation found by Oota and Yasui
[arXiv:0711.0078]. Since all the operators in the set can be generated from the
principal conformal Killing-Yano tensor, this establishes the (up to now)
missing link among the existence of hidden symmetry, presence of a complete set
of commuting operators, and separability of the Dirac equation in these
spacetimes.
