= Solution
For a free particle the <Dirac matrices> are constant and the momenta commute. Squaring the <Hamiltonian> cancels all mixed terms by their <anticommutators>:
$$
H_D^2=c^2\mathbf p^2+m^2c^4.
$$
Applying another <time> <derivative> to the <Dirac equation> therefore gives
$$
\boxed{\left(\frac1{c^2}\partial_t^2-\nabla^2+
\frac{m^2c^2}{\hbar^2}\right)\psi_\alpha=0}
$$
for every component. This is the <Klein-Gordon equation>. Its componentwise satisfaction is a necessary consequence of the first-order <Dirac equation>, not a converse selecting all spinor solutions of the squared equation.
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