U=(1/2) {{1,1,1,1},{1,I,-1,-I},{1,-1,1,-1},{1,-I,-1,I}};
U1={{((Conjugate[U[[1,1]]])/(Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])),((Conjugate[U[[2,1]]])/(Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])),0,0},{((U[[2,1]])/(Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])),((-U[[1,1]])/(Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])),0,0},{0,0,1,0},{0,0,0,1}};
U1U=U1.U;
U2={{((Conjugate[U1U[[1,1]]])/(Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])),0,((Conjugate[U1U[[3,1]]])/(Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])),0},{((U1U[[3,1]])/(Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])),0,((-U1U[[1,1]])/(Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])),0},{0,1,0,0},{0,0,0,1}};
U2U1U=Simplify[U2.U1U];
U3={{((Conjugate[U2U1U[[1,1]]])/(Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])),0,0,((Conjugate[U2U1U[[4,1]]])/(Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)]))},{((U2U1U[[4,1]])/(Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])),0,0,((-U2U1U[[1,1]])/(Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)]))},{0,1,0,0},{0,0,1,0}};
U3U2U1U=Simplify[U3.U2U1U];
U4={{1,0,0,0},{0,((Conjugate[U3U2U1U[[2,2]]])/(Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])),((Conjugate[U3U2U1U[[3,2]]])/(Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])),0},{0,((U3U2U1U[[3,2]])/(Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])),((-U3U2U1U[[2,2]])/(Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])),0},{0,0,0,1}};
U4U3U2U1U=Simplify[U4.U3U2U1U];
U5={{1,0,0,0},{0,Conjugate[U4U3U2U1U[[2,2]]]/(√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))),0,Conjugate[U4U3U2U1U[[4,2]]]/(√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2)))},{0,0,1,0},{0,U4U3U2U1U[[4,2]]/(√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))),0,(-U4U3U2U1U[[2,2]])/(√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2)))}};
U5U4U3U2U1U=Simplify[U5.U4U3U2U1U];
U6={{1,0,0,0},{0,1,0,0},{0,0,Conjugate[U5U4U3U2U1U[[3,3]]],Conjugate[U5U4U3U2U1U[[4,3]]]},{0,0,Conjugate[U5U4U3U2U1U[[3,4]]],Conjugate[U5U4U3U2U1U[[4,4]]]}};
U6U5U4U3U2U1U=Simplify[U6.U5U4U3U2U1U];
(U3U2U1U[[2;;,2;;]])//MatrixForm
Simplify[(ConjugateTranspose[U3U2U1U[[2;;,2;;]]].U3U2U1U[[2;;,2;;]])]//MatrixForm

U=(1/2) {{1,1,1,1},{1,I,-1,-I},{1,-1,1,-1},{1,-I,-1,I}};
U1={{((Conjugate[U[[1,1]]])/((((Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])))^2)),((Conjugate[U[[2,1]]])/((((Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])))^2)),0,0},{((U[[2,1]])/((((Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])))^2)),((-U[[1,1]])/((((Sqrt[((Abs[U[[1,1]]])^2)+((Abs[U[[2,1]]])^2)])))^2)),0,0},{0,0,1,0},{0,0,0,1}};
U1U=U1.U;
U2={{((Conjugate[U1U[[1,1]]])/((((Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])))^2)),0,((Conjugate[U1U[[3,1]]])/((((Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])))^2)),0},{((U1U[[3,1]])/((((Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])))^2)),0,((-U1U[[1,1]])/((((Sqrt[((Abs[U1U[[1,1]]])^2)+((Abs[U1U[[3,1]]])^2)])))^2)),0},{0,1,0,0},{0,0,0,1}};
U2U1U=Simplify[U2.U1U];
U3={{((Conjugate[U2U1U[[1,1]]])/((((Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])))^2)),0,0,((Conjugate[U2U1U[[4,1]]])/((((Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])))^2))},{((U2U1U[[4,1]])/((((Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])))^2)),0,0,((-U2U1U[[1,1]])/((((Sqrt[((Abs[U2U1U[[1,1]]])^2)+((Abs[U2U1U[[4,1]]])^2)])))^2))},{0,1,0,0},{0,0,1,0}};
U3U2U1U=Simplify[U3.U2U1U];
U4={{1,0,0,0},{0,((Conjugate[U3U2U1U[[2,2]]])/((((Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])))^2)),((Conjugate[U3U2U1U[[3,2]]])/((((Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])))^2)),0},{0,((U3U2U1U[[3,2]])/((((Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])))^2)),((-U3U2U1U[[2,2]])/((((Sqrt[((Abs[U3U2U1U[[2,2]]])^2)+((Abs[U3U2U1U[[3,2]]])^2)])))^2)),0},{0,0,0,1}};
U4U3U2U1U=Simplify[U4.U3U2U1U];
U5={{1,0,0,0},{0,Conjugate[U4U3U2U1U[[2,2]]]/(((√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))))^2),0,Conjugate[U4U3U2U1U[[4,2]]]/(((√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))))^2)},{0,0,1,0},{0,U4U3U2U1U[[4,2]]/(((√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))))^2),0,(-U4U3U2U1U[[2,2]])/(((√(((Abs[U4U3U2U1U[[2,2]]])^2)+((Abs[U4U3U2U1U[[4,2]]])^2))))^2)}};
U5U4U3U2U1U=Simplify[U5.U4U3U2U1U];
U6={{1,0,0,0},{0,1,0,0},{0,0,Conjugate[U5U4U3U2U1U[[3,3]]],Conjugate[U5U4U3U2U1U[[4,3]]]},{0,0,Conjugate[U5U4U3U2U1U[[3,4]]],Conjugate[U5U4U3U2U1U[[4,4]]]}};
U6U5U4U3U2U1U=Simplify[U6.U5U4U3U2U1U];
U6U5U4U3U2U1U//MatrixForm