  In this paper we derive the most general first-order symmetry operator
commuting with the Dirac operator in all dimensions and signatures. Such an
operator splits into Clifford even and Clifford odd parts which are given in
terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous
forms respectively. We study commutators of these symmetry operators and give
necessary and sufficient conditions under which they remain of the first-order.
In this specific setting we can introduce a Killing-Yano bracket, a bilinear
operation acting on odd Killing-Yano and even closed conformal Killing-Yano
forms, and demonstrate that it is closely related to the Schouten-Nijenhuis
bracket. An important non-trivial example of vanishing Killing-Yano brackets is
given by Dirac symmetry operators generated from the principal conformal
Killing-Yano tensor [hep-th/0612029]. We show that among these operators one
can find a complete subset of mutually commuting operators. These operators
underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all
dimensions [arXiv:0711.0078].
