The covariant form Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi(x)=0$ can be multiplied from the left with the operator $(i\gamma^\nu\partial_\nu+m)$ and 4xpanding it out, to get $$(i\gamma^\nu\partial_\nu+m)(i\gamma^\mu\partial_\mu-m)\psi(x)=0,\\ \Rightarrow(\gamma^\nu\gamma^\mu\partial_\nu\partial_\mu+m^2)\psi(x)=0.$$ Changing the indices $\mu\leftrightarrow\nu$, we also have an equation $$(\gamma^\mu\gamma^\nu\partial_\mu\partial_\nu+m^2)\psi(x)=0$$ Now if we add them and use the anticommutaror formula, $[\gamma^\mu,\gamma^\nu]_+=2\eta^{\mu\nu}$, we find, $$\big([\gamma^\mu,\gamma^\nu]_+\partial_\mu\partial_\nu+2m^2\big)\psi(x)=(2\eta^{\mu\nu}\partial_\mu\partial_\nu+2m^2)\psi(x)=0$$ which is same as $(\partial_\mu\partial^\mu+m^2)\psi(x)=0$ the Klein-Gordon equation.
Now clearly this must be wrong. A Dirac field is spin-$1/2$ fermion field and it cannot satisfy KG equation. But which step is wrong is derivation? If this derivation is correct, how should I interpret this? Should I interpret that each of the four components of $\psi(x)$ behaves like a KG field? Please help!
