  We show that the eigenspaces of the Dirac operator $H=\alpha\cdot (D - A(x))
+ m \beta $ at the threshold energies $\pm m$ are coincide with the direct sum
of the zero space and the kernel of the Weyl-Dirac operator $\sigma\cdot (D -
A(x))$. Based on this result, we describe the asymptotic limits of the
eigenfunctions of the Dirac operator corresponding to these threshold energies.
Also, we discuss the set of vector potentials for which the kernels of $H\mp m$
are non-trivial, i.e. ${Ker}(H\mp m) \not = \{0 \}$.
