= Solution
The supplied matrices are the <chiral gamma-matrix representation>. The <Pauli matrix multiplication law> implies $\{\sigma^i,\sigma^j\}=2\delta^{ij}I_2$. Block multiplication gives $(\gamma^0)^2=I_4$, $\{\gamma^0,\gamma^i\}=0$, and $\{\gamma^i,\gamma^j\}=-2\delta^{ij}I_4$. Hence their <Clifford algebra> relation is
$$
\boxed{\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}I_4.}
$$
Since partial derivatives commute, multiplying the <Dirac equation> by $i\gamma^\nu\partial_\nu+m$ gives
$$
0=(i\gamma^\nu\partial_\nu+m)(i\gamma^\mu\partial_\mu-m)\psi
=-(g^{\mu\nu}\partial_\mu\partial_\nu+m^2)\psi.
$$
Thus every component of the <Dirac spinor> satisfies $\boxed{(\Box+m^2)\psi=0}$, the <Klein-Gordon equation> with the same mass.
For a <Lorentz transformation> $x'=\Lambda x$, set $\psi'(x')=S(\Lambda)\psi(x)$. The derivative transforms as $\partial'_\mu=(\Lambda^{-1})^\nu{}_{\mu}\partial_\nu$. With the given spinor-matrix identity,
$$
S^{-1}(i\gamma^\mu\partial'_\mu-m)\psi'
=[i\Lambda^\mu{}_{\rho}\gamma^\rho(\Lambda^{-1})^\nu{}_{\mu}\partial_\nu-m]\psi
=(i\gamma^\nu\partial_\nu-m)\psi=0.
$$
This proves the <Lorentz covariance of the Dirac operator> explicitly.
For <parity>, let $x_P=(x^0,-\mathbf x)$ and $\psi_P(x)=\gamma^0\psi(x_P)$. Spatial differentiation brings a minus sign, which is canceled by $\gamma^i\gamma^0=-\gamma^0\gamma^i$. Therefore
$$
(i\gamma^\mu\partial_\mu-m)\psi_P(x)
=\gamma^0[(i\gamma^\mu\partial_\mu-m)\psi](x_P)=0.
$$
Since $\gamma^0$ is Hermitian and squares to one, the <Dirac adjoint> transforms as
$$
\boxed{\bar\psi_P(x)=\bar\psi(x_P)\gamma^0.}
$$
The <chirality matrix> $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ anticommutes with $\gamma^0$. The <parity transformation of a Dirac bilinear> consequently gives
$$
\boxed{\bar\psi_P\psi_P(x)=\bar\psi\psi(x_P),\qquad
\bar\psi_P\gamma^5\psi_P(x)=-\bar\psi\gamma^5\psi(x_P).}
$$
The first bilinear is a <Lorentz scalar> even under <parity>, and the second is a <pseudoscalar>.
In this representation, $\gamma^5=\operatorname{diag}(-I_2,I_2)$. Its square is one, so the <chiral projectors> $P_\pm=(1\pm\gamma^5)/2$ are idempotent, mutually orthogonal and sum to the identity. They select $\psi_L=P_-\psi$ and $\psi_R=P_+\psi$, the two <Weyl spinors>. Because $\gamma^\mu P_\pm=P_\mp\gamma^\mu$, a solution of the <massless Dirac equation> has each chiral component solving that equation separately. In two-component form,
$$
i(\partial_t-\boldsymbol\sigma\cdot\nabla)\psi_L=0,\qquad
i(\partial_t+\boldsymbol\sigma\cdot\nabla)\psi_R=0.
$$
The two <chirality> sectors are thus independent under massless propagation and proper <Lorentz transformations>. A nonzero <Dirac mass term> couples them, while <parity> exchanges them. This explains why the projectors are particularly useful in the massless theory.
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