= Solution
Use the metric in the question and fix the <Levi-Civita symbol> convention $\epsilon_{0123}=+1$. A consistent choice of rotation <Lie algebra generators> is
$$
\boxed{J_1=M^{23},\qquad J_2=M^{31},\qquad J_3=M^{12}.}
$$
Substituting the indices into the printed <Poincare algebra> gives
$$
[J_1,J_2]=[M^{23},M^{31}]=\eta^{33}M^{21}
=-M^{21}=M^{12}=J_3.
$$
The other cyclic brackets follow the same way. The negative spatial metric and antisymmetry of $M^{\rho\sigma}$ both enter this sign.
The printed brackets are real <Lie algebra> brackets, without the factor $i$ used for ordinary <commutators> of Hermitian quantum observables. We use those brackets for the algebraic verifications. For physical eigenvalues below, angular momentum is Hermitian, its <spin> projection is the real number $j_3$, and $\hbar=1$. In that quantum convention the ordinary operator <commutators> are $i$ times the displayed brackets. This distinction prevents identifying a real <spin> projection with an anti-Hermitian <matrix> eigenvalue.
In the <universal enveloping algebra>, translations commute. The <Pauli-Lubanski pseudovector> therefore satisfies
$$
W_\mu P^\mu=\frac12\epsilon_{\mu\nu\rho\tau}M^{\nu\rho}P^\tau P^\mu=0.
$$
For fixed $\nu,\rho$, the product of momenta is symmetric in $\mu,\tau$ while the epsilon coefficient is antisymmetric in them. There is no need to commute the Lorentz generator through the momenta.
Using the <commutator derivation identity> and the mixed bracket,
$$
\begin{aligned}
[W_\mu,P^\sigma]
&=\frac12\epsilon_{\mu\nu\rho\tau}
\bigl(\eta^{\rho\sigma}P^\nu-\eta^{\nu\sigma}P^\rho\bigr)P^\tau\\
&=\epsilon_{\mu\nu\rho\tau}\eta^{\rho\sigma}P^\nu P^\tau=0.
\end{aligned}
$$
The two terms become equal after swapping $\nu,\rho$, and the last expression vanishes by antisymmetry in $\nu,\tau$. In the Hermitian observable convention, the calculation has one overall extra $i$ and still vanishes. Thus \b[$W_\mu$ preserves each momentum eigenspace].
For the specified epsilon orientation, two useful component identities are
$$
W_0=J_1P^1+J_2P^2+J_3P^3,
\qquad W_3=-J_3P^0+M^{02}P^1-M^{01}P^2.
$$
The order displayed matters: the momentum operator is on the right, so it can act first on the momentum eigenstate. At rest, $p^\mu=(m,0,0,0)$, and on a <spin> state with $J_3|j,j_3\rangle=j_3|j,j_3\rangle$ these give
$$
\boxed{W_0|\psi\rangle=0,\qquad W_3|\psi\rangle=-mj_3|\psi\rangle.}
$$
These are <massive rest-frame Pauli-Lubanski eigenvalues>. The raised component would be $W^3=+mj_3$; confusing $W_3$ with $W^3$ reverses the answer.
<Helicity> is the projection of <spin angular momentum>, or equivalently the rotation generator acting internally, along the momentum direction:
$$
h=\frac{\mathbf J\cdot\mathbf p}{|\mathbf p|}.
$$
For the momentum $p^\mu=(k,0,0,k)$ with $k>0$, the <helicity> operator is $J_3$. Hence a <helicity>-$j_3$ state has
$$
\boxed{W_0|\psi\rangle=kj_3|\psi\rangle,\qquad
W_3|\psi\rangle=-kj_3|\psi\rangle.}
$$
The result uses only the two longitudinal components and does not need a separate assumption about the transverse little-group generators. These <massless longitudinal Pauli-Lubanski eigenvalues> agree with $W^\mu=j_3P^\mu$ for ordinary finite-<helicity> representations.
Finally, at rest the contraction is $mW_0=0$. For the chosen null momentum it is $k(W_0+W_3)$, whose two eigenvalues cancel. Thus \b[both results obey $W_\mu P^\mu=0$]. Reversing the epsilon orientation reverses all $W$ eigenvalues together; it leaves both identities and both consistency checks intact. For a literal anti-Hermitian derived representation $\rho$ of the printed real algebra, write $H_X=i\rho(X)$ for the Hermitian observable. Since $W_\mu$ is bilinear in generators, its abstract enveloping-algebra image is $\rho(W_\mu)=-W_{\mu,\mathrm{phys}}$. Thus the corresponding formal-image eigenvalues, if that convention is intended, are
$$
\boxed{\text{rest}:\ (\rho(W_0),\rho(W_3))=(0,+mj_3),\qquad
\text{null}:\ (\rho(W_0),\rho(W_3))=(-kj_3,+kj_3).}
$$
Here $\rho(P^\mu)$ has eigenvalue $-ip^\mu$, while $p^\mu$ and $j_3$ themselves remain real physical labels. This is the same result after the <Hermitian quantum generator convention> is applied to both factors, not an inconsistent choice of <spin> sign. Both the Hermitian-observable and literal anti-Hermitian interpretations are consequently specified.
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