 Normalization of a Pauli-Lubanski spin direction

The physical massive-particle Pauli-Lubanski pseudovector obeys $W\cdot P=0$ and $W^2=-m^2{c}^2|{\mathbf{s}}_{\mathrm{rest}}{|}^2$ for classical rest-frame spin. Orthogonality by itself does not establish this normalization. In particular the components $W=(\mathbf{s}\cdot \mathbf{p},(E/c)\mathbf{s})$ give $W^2=-m^2{c}^2|\mathbf{s}{|}^2-|\mathbf{s}\times \mathbf{p}{|}^2$; dividing by $mc|\mathbf{s}|$ is a unit spacelike normalization only when the cross product vanishes.