We consider the boost Lz(b) | p_x, μ> = Lz(b) R(π/2) Lz(η) | 0, μ>. We write the composition of boost and rotations

(({{Cosh[b], 0, 0, Sinh[b]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sinh[b], 0, 0, Cosh[b]}})).(({{1, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}, {0, -1, 0, 0}})).(({{Cosh[η], 0, 0, Sinh[η]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sinh[η], 0, 0, Cosh[η]}}))//FullSimplify//MatrixForm

We compare with the composition R(θ’) Lz(η’). It doesn’t work (compare the position of zeroes)

(({{1, 0, 0, 0}, {0, Cos[θp], 0, Sin[θp]}, {0, 0, 1, 0}, {0, -Sin[θp], 0, Cos[θp]}})).(({{Cosh[ηp], 0, 0, Sinh[ηp]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sinh[ηp], 0, 0, Cosh[ηp]}}))//FullSimplify//MatrixForm

We introduce the Wigner rotation R(θ’) Lz(η’) R(ω). We can determine the values of θp, ηp and ω to match with the other one

(({{1, 0, 0, 0}, {0, Cos[θp], 0, Sin[θp]}, {0, 0, 1, 0}, {0, -Sin[θp], 0, Cos[θp]}})).(({{Cosh[ηp], 0, 0, Sinh[ηp]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sinh[ηp], 0, 0, Cosh[ηp]}})).(({{1, 0, 0, 0}, {0, Cos[ω], 0, Sin[ω]}, {0, 0, 1, 0}, {0, -Sin[ω], 0, Cos[ω]}}))//FullSimplify//MatrixForm

((({{1, 0, 0, 0}, {0, Cos[θp], 0, Sin[θp]}, {0, 0, 1, 0}, {0, -Sin[θp], 0, Cos[θp]}})).(({{Cosh[ηp], 0, 0, Sinh[ηp]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sinh[ηp], 0, 0, Cosh[ηp]}})).(({{1, 0, 0, 0}, {0, Cos[ω], 0, Sin[ω]}, {0, 0, 1, 0}, {0, -Sin[ω], 0, Cos[ω]}}))//FullSimplify)/.{Cosh[ηp]->Cosh[b] Cosh[η],Sinh[ηp]->(Sqrt[((Cosh[b])^2) ((Cosh[η])^2)-1]),Sin[θp]->Sinh[η]/(Sqrt[((Cosh[b])^2) ((Cosh[η])^2)-1]),Cos[θp]->Cosh[η] Sinh[b]/(Sqrt[((Cosh[b])^2) ((Cosh[η])^2)-1]) ,Sin[ω]->Sinh[b]/(Sqrt[((Cosh[b])^2) ((Cosh[η])^2)-1]),Cos[ω]-> Cosh[b] Sinh[η]/(Sqrt[((Cosh[b])^2) ((Cosh[η])^2)-1]) }//FullSimplify//MatrixForm