How to show straightforvard (i.e., via definition and Poincare's algebra), that $$ [\hat{W}_{\alpha}, \hat {J}_{\mu \nu}] = i\left( g_{\alpha \mu}\hat {W}_{\nu} - g_{\alpha \nu}\hat {W}_{\mu}\right), $$ for $$ [\hat {P}_{\alpha}, \hat {P}_{\beta }] = 0, \quad [\hat {P}_{\alpha}, \hat {J}_{\beta \gamma } ] = i(g_{\alpha \beta} \hat {P}_{\gamma} - g_{\alpha \gamma}\hat {P}_{\beta}), \quad [\hat {J}_{\alpha \beta}, \hat {J}_{\gamma \delta } ] = i\left( -g_{\alpha \delta }\hat {J}_{\gamma \beta } + g_{\beta \gamma}\hat {J}_{\alpha \delta} + g_{\alpha \gamma}\hat {J}_{\delta \beta } - g_{\beta \delta }\hat {J}_{\alpha \gamma}\right) $$ and $$ \hat {W}^{\alpha} = \varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}\hat {P}_{\delta}, \quad \varepsilon^{0123} = 1, \quad \varepsilon_{0123} = -1 ? $$ I tried to show than, but only got $$ [\hat {J}_{ij}, \hat {W}_{\alpha}] = \varepsilon^{\alpha \beta \gamma \delta}[\hat {J}_{ij}, \hat {J}_{\beta \gamma}\hat {P}_{\delta}] = \varepsilon^{\alpha \beta \gamma \delta}[\hat {J}_{ij}, \hat {J}_{\beta \gamma}]\hat {P}_{\delta } + \varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}[\hat {J}_{ij}, \hat {P}_{\delta }] = $$ $$ i\varepsilon^{\alpha \beta \gamma \delta}\left( -g_{i\gamma}\hat {J}_{\beta j}\hat {P}_{\delta } + g_{j\beta }\hat {J}_{i \gamma}\hat {P}_{\delta } + g_{i\beta }\hat {J}_{\gamma j}\hat {P}_{\delta } - g_{j\gamma }\hat{J}_{i \beta }\hat {P}_{\delta } + g_{\delta i}\hat {J}_{\beta \gamma}\hat {P}_{j} - g_{\delta j}\hat {J}_{\beta \gamma}\hat {P}_{i}\right) . $$ What can I do next?
Addition. I did it!!! But not by the method above.
I made some hint: it's easy to show, that $$ \hat {W}^{\mu}\hat {P}_{\mu} = \frac{1}{2}\varepsilon^{\mu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\hat {P}_{\mu} = 0 $$ as the convolution of symmetrical $\hat {P}_{\gamma }\hat {P}_{\mu}$ and antisymmetrical $\varepsilon^{\mu \alpha \beta \gamma}$.
So the commutator $$ [\hat {J}_{\kappa \lambda}, \hat {W}^{\mu}\hat {P}_{\mu}] = 0. \qquad (.1) $$ But by the other hand $$ [ \hat {J}_{\kappa \lambda}, \hat {W}^{\mu}\hat {P}_{\mu}] = \hat {W}^{\mu}[ \hat {J}_{\kappa \lambda}, \hat {P}_{\mu}] + [\hat {J}_{\kappa \lambda }, \hat {W}^{\mu}]\hat {P}_{\mu} = -\hat {W}^{\mu}i(g_{\mu \kappa }\hat {P}_{\lambda} - g_{\mu \lambda}\hat {P}_{\kappa }) + [\hat {J}_{\kappa \lambda }, \hat {W}^{\mu}]\hat {P}_{\mu} . $$ So, by using $(.1)$ from the previous formula there follows the next: $$ [\hat {J}_{\kappa \lambda }, \hat {W}^{\mu}]\hat {P}_{\mu} = i\hat {W}^{\mu}(g_{\mu \kappa }\hat {P}_{\lambda} - g_{\mu \lambda}\hat {P}_{\kappa }) = i(\hat {W}_{\kappa}\hat {P}_{\lambda} - \hat {W}_{\lambda}\hat {P}_{\kappa}) = i\left(\hat {W}_{\kappa}\delta^{\mu}_{\lambda} - \hat {W}_{\lambda}\delta^{\mu}_{\kappa}\right){P}_{\mu}, $$ where $\delta^{0}_{0} = 1, \delta^{i}_{i} = -1$,
and, finally, $$ [\hat {J}_{\kappa \lambda }, \hat {W}^{\mu}] = i\left(\hat {W}_{\kappa}\delta^{\nu}_{\lambda} - \hat {W}_{\lambda}\delta^{\mu}_{\kappa}\right) \Rightarrow [\hat {J}_{\kappa \lambda }, \hat {W}_{\mu}] = i\left(\hat {W}_{\kappa}g_{\mu \lambda} - \hat {W}_{\lambda}g_{\mu \kappa}\right). $$
But the question is still actual.